## Ejercicio 110

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## Ejercicio 109

CAPITULO X

Descomposición Factorial
Ejercicio 109
Descomponer en cinco factores:
1. $\begin{array}{cc}{x}^{9}–x{y}^{8}& =x\left({x}^{8}–{y}^{8}\right)\\ & =x\left({x}^{4}+{y}^{4}\right)\left({x}^{4}–{y}^{4}\right)\\ & =x\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left({x}^{2}–{y}^{2}\right)\\ & =x\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left(x+y\right)\left(x–y\right)\end{array}$
2. $\begin{array}{cccc}{x}^{5}–40{x}^{3}+144x& =x\left({x}^{4}–40{x}^{2}+144\right)& & .\begin{array}{c}144\\ 72\\ 36\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 36\\ \end{array}\\ & =x\left({x}^{4}–36{x}^{2}–4{x}^{2}+144\right)& & \\ & =x\left[{x}^{2}\left({x}^{2}–36\right)–4\left({x}^{2}–36\right)\right]& & \\ & =x\left({x}^{2}–36\right)\left({x}^{2}–4\right)& & \\ & =x\left(x+6\right)\left(x–6\right)\left(x+2\right)\left(x–2\right)& & \end{array}$
3. $\begin{array}{cc}{a}^{6}+{a}^{3}{b}^{3}–{a}^{4}–a{b}^{3}& =a\left[{a}^{5}+{a}^{2}{b}^{3}–{a}^{3}–{b}^{3}\right]\\ & =a\left[{a}^{2}\left({a}^{3}+{b}^{3}\right)–\left({a}^{3}+{b}^{3}\right)\right]\\ & =a\left({a}^{3}+{b}^{3}\right)\left({a}^{2}–1\right)\\ & =a\left(a+b\right)\left({a}^{2}–ab+{b}^{2}\right)\left(a+1\right)\left(a–1\right)\end{array}$
4. $\begin{array}{cc}4{x}^{4}–8{x}^{2}+4& =4\left({x}^{4}–2{x}^{2}+1\right)\\ & =4{\left({x}^{2}–1\right)}^{2}\\ & =4{\left[\left(x–1\right)\left(x+1\right)\right]}^{2}\\ & =4{\left(x–1\right)}^{2}{\left(x+1\right)}^{2}\\ & =4\left(x–1\right)\left(x–1\right)\left(x+1\right)\left(x+1\right)\end{array}$
5. $\begin{array}{cc}{a}^{7}–a{b}^{6}& =a\left({a}^{6}–{b}^{6}\right)\\ & =a\left({a}^{3}+{b}^{3}\right)\left({a}^{3}–{b}^{3}\right)\\ & =a\left(a+b\right)\left({a}^{2}–ab+{b}^{2}\right)\left(a–b\right)\left({a}^{2}+ab+{b}^{2}\right)\end{array}$
6. $\begin{array}{cc}2{a}^{4}–2{a}^{3}–4{a}^{2}–2{a}^{2}{b}^{2}+2a{b}^{2}+4{b}^{2}& =2{a}^{2}\left({a}^{2}–a–2\right)–2{b}^{2}\left({a}^{2}–a–2\right)\\ & =\left({a}^{2}–a–2\right)\left(2{a}^{2}–2{b}^{2}\right)\\ & =a\left({a}^{2}–{b}^{2}\right)\left({a}^{2}–a–2\right)\\ & =a\left({a}^{2}–{b}^{2}\right)\left({a}^{2}–2a+a–2\right)\\ & =a\left(a+b\right)\left(a–b\right)\left[a\left(a–2\right)+\left(a–2\right)\right]\\ & =a\left(a+b\right)\left(a–b\right)\left(a–2\right)\left(a+1\right)\end{array}$
7. $\begin{array}{cc}{x}^{6}+5{x}^{5}–81{x}^{2}–405x& ={x}^{6}–81{x}^{2}+5{x}^{5}–405x\\ & ={x}^{2}\left({x}^{4}–81\right)+5x\left({x}^{4}–81\right)\\ & =\left({x}^{4}–81\right)\left({x}^{2}+5x\right)\\ & =x\left({x}^{2}+9\right)\left({x}^{2}–9\right)\left(x+5\right)\\ & =x\left({x}^{2}+9\right)\left(x+3\right)\left(x–3\right)\left(x+5\right)\end{array}$
8. $\begin{array}{cc}3–3{a}^{6}& =3\left(1–{a}^{6}\right)\\ & =3\left(1–{a}^{3}\right)\left(1+{a}^{3}\right)\\ & =3\left(1–a\right)\left(1+a+{a}^{2}\right)\left(1+a\right)\left(1–a+{a}^{2}\right)\end{array}$
9. $\begin{array}{cc}4a{x}^{2}\left({a}^{2}–2ax+{x}^{2}\right)–{a}^{3}+2{a}^{2}x–a{x}^{2}& =4a{x}^{2}{\left(a–x\right)}^{2}–a\left({a}^{2}–2ax+{x}^{2}\right)\\ & =4a{x}^{2}{\left(a–x\right)}^{2}–a{\left(a–x\right)}^{2}\\ & ={\left(a–x\right)}^{2}\left[4a{x}^{2}–a\right]\\ & =a{\left(a–x\right)}^{2}\left[4{x}^{2}–1\right]\\ & =a\left(a–x\right)\left(a–x\right)\left(2x–1\right)\left(2x+1\right)\end{array}$
10. $\begin{array}{cc}{x}^{7}+{x}^{4}–81{x}^{3}–81& ={x}^{4}\left({x}^{3}+1\right)–81\left({x}^{3}+1\right)\\ & =\left({x}^{3}+1\right)\left({x}^{4}–81\right)\\ & =\left(x+1\right)\left({x}^{2}–x+1\right)\left({x}^{2}+9\right)\left({x}^{2}–9\right)\\ & =\left(x+1\right)\left({x}^{2}–x+1\right)\left({x}^{2}+9\right)\left(x+3\right)\left(x–3\right)\end{array}$
Descomponer en seis factores
11. $\begin{array}{cc}{x}^{17}–x& =x\left({x}^{16}–1\right)\\ & =x\left({x}^{8}+1\right)\left({x}^{8}–1\right)\\ & =x\left({x}^{8}+1\right)\left({x}^{4}+1\right)\left({x}^{4}–1\right)\\ & =x\left({x}^{8}+1\right)\left({x}^{4}+1\right)\left({x}^{2}+1\right)\left({x}^{2}–1\right)\\ & =x\left({x}^{8}+1\right)\left({x}^{4}+1\right)\left({x}^{2}+1\right)\left(x+1\right)\left(x–1\right)\end{array}$
12. $\begin{array}{cc}3{x}^{6}–75{x}^{4}–48{x}^{2}+1200& =3{x}^{6}–48{x}^{2}–75{x}^{4}+1200\\ & =3{x}^{2}\left({x}^{4}–16\right)–75\left({x}^{4}–16\right)\\ & =\left({x}^{4}–16\right)\left(3{x}^{2}–75\right)\\ & =3\left({x}^{2}+4\right)\left({x}^{2}–4\right)\left({x}^{2}–25\right)\\ & =3\left({x}^{2}+4\right)\left(x+2\right)\left(x–2\right)\left(x+5\right)\left(x–5\right)\end{array}$
13. $\begin{array}{cc}{a}^{6}{x}^{2}–{x}^{2}+{a}^{6}x–x& ={x}^{2}\left({a}^{6}–1\right)+x\left({a}^{6}–1\right)\\ & =\left({a}^{6}–1\right)\left({x}^{2}+x\right)\\ & =x\left({a}^{3}+1\right)\left({a}^{3}–1\right)\left(x+1\right)\\ & =x\left(a+1\right)\left({a}^{2}–a+1\right)\left(a–1\right)\left({a}^{2}+a+1\right)\left(x+1\right)\end{array}$
14. $\begin{array}{cccc}\left({a}^{2}–ax\right)\left({x}^{4}–82{x}^{2}+81\right)& =a\left(a–x\right)\left({x}^{4}–{x}^{2}–81{x}^{2}+81\right)& & .\begin{array}{c}81\\ 1\end{array}|\begin{array}{c}81\\ \end{array}\\ & =a\left(a–x\right)\left[{x}^{2}\left({x}^{2}–1\right)–81\left({x}^{2}–1\right)\right]& & \\ & =a\left(a–x\right)\left({x}^{2}–1\right)\left({x}^{2}–81\right)& & \\ & =a\left(a–x\right)\left(x+1\right)\left(x–1\right)\left(x+9\right)\left(x–9\right)& & \end{array}$

## Ejercicio 108

CAPITULO X

Descomposición Factorial
Ejercicio 108
Descomponer en cuatro factores:
1. $\begin{array}{cc}1–{a}^{8}& =\left(1+{a}^{4}\right)\left(1–{a}^{4}\right)\\ & =\left(1+{a}^{4}\right)\left(1+{a}^{2}\right)\left(1–{a}^{2}\right)\\ & =\left(1+{a}^{4}\right)\left(1+{a}^{2}\right)\left(1+a\right)\left(1–a\right)\end{array}$
2. $\begin{array}{cc}{a}^{6}–1& =\left({a}^{3}–1\right)\left({a}^{3}+1\right)\\ & =\left(a–1\right)\left({a}^{2}+a+1\right)\left(a+1\right)\left({a}^{2}–a+1\right)\end{array}$
3. $\begin{array}{cc}{x}^{4}–41{x}^{2}+400& ={x}^{4}–41{x}^{2}+400+{x}^{2}–{x}^{2}\\ & ={x}^{4}–40{x}^{2}+400–{x}^{2}\\ & ={\left({x}^{2}–20\right)}^{2}–{x}^{2}\\ & =\left[\left({x}^{2}–20\right)–x\right]\left[\left({x}^{2}–20\right)+x\right]\\ & =\left({x}^{2}–x–20\right)\left({x}^{2}+x–20\right)\\ & =\left({x}^{2}–5x+4x–20\right)\left({x}^{2}–4x+5x–20\right)\\ & =\left[x\left(x–5\right)+4\left(x–5\right)\right]\left[x\left(x–4\right)+5\left(x–4\right)\right]\\ & =\left(x–5\right)\left(x+4\right)\left(x–4\right)\left(x+5\right)\end{array}$
4. $\begin{array}{cc}{a}^{4}–2{a}^{2}{b}^{2}+{b}^{4}& ={\left({a}^{2}–{b}^{2}\right)}^{2}\\ & ={\left[\left(a–b\right)\left(a+b\right)\right]}^{2}\\ & ={\left(a–b\right)}^{2}{\left(a+b\right)}^{2}\\ & =\left(a–b\right)\left(a–b\right)\left(a+b\right)\left(a+b\right)\end{array}$
5. $\begin{array}{cccc}{x}^{5}+{x}^{3}–2x& =x\left({x}^{4}+{x}^{2}–2\right)& & .\begin{array}{c}2\\ 1\end{array}|\begin{array}{c}2\\ \end{array}\\ & =x\left({x}^{4}–{x}^{2}+2{x}^{2}–2\right)& & \\ & =x\left[{x}^{2}\left({x}^{2}–1\right)+\left({x}^{2}–1\right)\right]& & \\ & =x\left({x}^{2}–1\right)\left({x}^{2}+1\right)& & \\ & =x\left(x–1\right)\left(x+1\right)\left({x}^{2}+1\right)& & \end{array}$
6. $\begin{array}{cc}2{x}^{4}+6{x}^{3}–2x–6& =2{x}^{3}\left(x+3\right)–2\left(x+3\right)\\ & =\left(x+3\right)\left(2{x}^{3}–2\right)\\ & =2\left(x+3\right)\left({x}^{3}–1\right)\\ & =2\left(x+3\right)\left(x–1\right)\left({x}^{2}+x+1\right)\end{array}$
7. $\begin{array}{cc}3{x}^{4}–243& =3\left({x}^{4}–81\right)\\ & =3\left({x}^{2}+9\right)\left({x}^{2}–9\right)\\ & =3\left({x}^{2}+9\right)\left(x–3\right)\left(x+3\right)\end{array}$
8. $\begin{array}{cc}16{x}^{4}–8{x}^{2}{y}^{2}+{y}^{4}& ={\left(4{x}^{2}–{y}^{2}\right)}^{2}\\ & ={\left[\left(2x+y\right)\left(2x–y\right)\right]}^{2}\\ & ={\left(2x+y\right)}^{2}{\left(2x–y\right)}^{2}\\ & =\left(2x+y\right)\left(2x+y\right)\left(2x–y\right)\left(2x–y\right)\end{array}$
9. $\begin{array}{cc}9{x}^{4}+9{x}^{3}y–{x}^{2}–xy& =9{x}^{3}\left(x+y\right)–x\left(x+y\right)\\ & =\left(x+y\right)\left(9{x}^{3}–x\right)\\ & =x\left(x+y\right)\left(9{x}^{2}–1\right)\\ & =x\left(x+y\right)\left(3x–1\right)\left(3x+1\right)\end{array}$
10. $\begin{array}{cccc}12a{x}^{4}+33a{x}^{2}–9a& =3a\left(4{x}^{4}+11{x}^{2}–3\right)& & .\begin{array}{c}12\\ 1\end{array}|\begin{array}{c}12\\ \end{array}\\ & =3a\left(4{x}^{4}–{x}^{2}+12{x}^{2}–3\right)& & \\ & =3a\left[{x}^{2}\left(4{x}^{2}–1\right)+3\left(4{x}^{2}–1\right)\right]& & \\ & =3a\left({x}^{2}+3\right)\left(4{x}^{2}–1\right)& & \\ & =3a\left({x}^{2}+3\right)\left(2x–1\right)\left(2x+1\right)& & \end{array}$
11. $\begin{array}{cc}{x}^{8}–{y}^{8}& =\left({x}^{4}+{y}^{4}\right)\left({x}^{4}–{y}^{4}\right)\\ & =\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left({x}^{2}–{y}^{2}\right)\\ & =\left({x}^{4}+{y}^{4}\right)\left({x}^{2}+{y}^{2}\right)\left(x+y\right)\left(x–y\right)\end{array}$
12. $\begin{array}{cccc}{x}^{6}–7{x}^{3}–8& ={x}^{6}–8{x}^{3}+{x}^{3}–8& & .\begin{array}{c}8\\ 1\end{array}|\begin{array}{c}8\\ \end{array}\\ & ={x}^{3}\left({x}^{3}–8\right)+\left({x}^{3}–8\right)& & \\ & =\left({x}^{3}–8\right)\left({x}^{3}+8\right)& & \\ & =\left(x–2\right)\left({x}^{2}+2x+4\right)\left(x+2\right)\left({x}^{2}–2x+4\right)& & \end{array}$
13. $\begin{array}{cc}64–{x}^{6}& =\left(8–{x}^{3}\right)\left(8+{x}^{3}\right)\\ & =\left(2–x\right)\left(4+2x+{x}^{2}\right)\left(2+x\right)\left(4–2x+{x}^{2}\right)\end{array}$
14. $\begin{array}{cc}{a}^{5}–{a}^{3}{b}^{2}–{a}^{2}{b}^{3}+{b}^{5}& ={a}^{3}\left({a}^{2}–{b}^{2}\right)–{b}^{3}\left({a}^{2}–{b}^{2}\right)\\ & =\left({a}^{2}–{b}^{2}\right)\left({a}^{3}–{b}^{3}\right)\\ & =\left(a–b\right)\left(a+b\right)\left(a–b\right)\left({a}^{2}+ab+{b}^{2}\right)\end{array}$
15. $\begin{array}{cccc}8{x}^{4}+6{x}^{2}–2& =2\left(4{x}^{4}+3{x}^{2}–1\right)& & .\begin{array}{c}4\\ 1\end{array}|\begin{array}{c}4\\ \end{array}\\ & =2\left(4{x}^{4}–{x}^{2}+4{x}^{2}–1\right)& & \\ & =2\left[{x}^{2}\left(4{x}^{2}–1\right)+\left(4{x}^{2}–1\right)\right]& & \\ & =2\left(4{x}^{2}–1\right)\left({x}^{2}+1\right)& & \\ & =2\left(2x+1\right)\left(2x–1\right)\left({x}^{2}+1\right)& & \end{array}$
16. $\begin{array}{cccc}{a}^{4}–25{a}^{2}+144& ={a}^{4}–9{a}^{2}–16{a}^{2}+144& & .\begin{array}{c}144\\ 72\\ 36\\ 18\\ 9\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 2\\ 9\\ \end{array}\\ & ={a}^{2}\left({a}^{2}–9\right)–16\left({a}^{2}–9\right)& & \\ & =\left({a}^{2}–9\right)\left({a}^{2}–16\right)& & \\ & =\left(a–9\right)\left(a+3\right)\left(a+4\right)\left(a–4\right)& & \end{array}$
17. $\begin{array}{cc}{a}^{2}{x}^{3}–{a}^{2}{y}^{3}+2a{x}^{3}–2a{y}^{3}& ={a}^{2}\left({x}^{3}–{y}^{3}\right)+2a\left({x}^{3}–{y}^{3}\right)\\ & =\left({a}^{2}+2a\right)\left({x}^{3}–{y}^{3}\right)\\ & =a\left(a+2\right)\left(x–y\right)\left({x}^{2}+xy+{y}^{2}\right)\end{array}$
18. $\begin{array}{cc}{a}^{4}+2{a}^{3}–{a}^{2}–2a& ={a}^{4}–{a}^{2}+2{a}^{3}–2a\\ & ={a}^{2}\left({a}^{2}–1\right)+2a\left({a}^{2}–1\right)\\ & =\left({a}^{2}+2a\right)\left({a}^{2}–1\right)\\ & =a\left(a+2\right)\left(a–1\right)\left(a+1\right)\end{array}$
19. $\begin{array}{cc}1–2{a}^{3}+{a}^{6}& ={\left(1–{a}^{3}\right)}^{2}\\ & ={\left[\left(1–a\right)\left(1+a+{a}^{2}\right)\right]}^{2}\\ & ={\left(1–a\right)}^{2}{\left(1+a+{a}^{2}\right)}^{2}\end{array}$
20. $\begin{array}{cc}{m}^{6}–729& =\left({m}^{3}+27\right)\left({m}^{3}–27\right)\\ & =\left(m+3\right)\left({m}^{2}–3m+9\right)\left(m–3\right)\left({m}^{2}+3m+9\right)\end{array}$
21. $\begin{array}{cc}{x}^{5}–x& =x\left({x}^{4}–1\right)\\ & =x\left({x}^{2}+1\right)\left({x}^{2}–1\right)\\ & =x\left({x}^{2}+1\right)\left(x–1\right)\left(x+1\right)\end{array}$
22. $\begin{array}{cc}{x}^{5}–{x}^{3}{y}^{2}+{x}^{2}{y}^{3}–{y}^{5}& ={x}^{3}\left({x}^{2}–{y}^{2}\right)+{y}^{3}\left({x}^{2}–{y}^{2}\right)\\ & =\left({x}^{2}–{y}^{2}\right)\left({x}^{3}+{y}^{3}\right)\\ & =\left(x+y\right)\left(x–y\right)\left(x+y\right)\left({x}^{2}–xy+{y}^{2}\right)\end{array}$
23. $\begin{array}{cc}{a}^{4}b–{a}^{3}{b}^{2}–{a}^{2}{b}^{3}+a{b}^{4}& ={a}^{3}b\left(a–b\right)–a{b}^{3}\left(a–b\right)\\ & =\left(a–b\right)\left({a}^{3}b–a{b}^{3}\right)\\ & =ab\left(a–b\right)\left({a}^{2}–{b}^{2}\right)\\ & =ab\left(a–b\right)\left(a+b\right)\left(a–b\right)\end{array}$
24. $\begin{array}{cc}5{a}^{4}–3125& =5\left({a}^{4}–625\right)\\ & =5\left({a}^{2}+25\right)\left({a}^{2}–25\right)\\ & =5\left({a}^{2}+25\right)\left(a–5\right)\left(a+5\right)\end{array}$
25. $\begin{array}{cccc}{\left({a}^{2}+2a\right)}^{2}–2\left({a}^{2}+2a\right)–3& ={\left({a}^{2}+2a\right)}^{2}–3\left({a}^{2}+2a\right)+\left({a}^{2}+2a\right)–3& & .\begin{array}{c}3\\ 1\end{array}|\begin{array}{c}3\\ \end{array}\\ & =\left({a}^{2}+2a\right)\left[\left({a}^{2}+2a\right)–3\right]+\left[\left({a}^{2}+2a\right)–3\right]& & \\ & =\left[\left({a}^{2}+2a\right)–3\right]\left[\left({a}^{2}+2a\right)+1\right]& & \\ & =\left({a}^{2}+2a–3\right)\left({a}^{2}+2a+1\right)& & \\ & =\left({a}^{2}–a+3a–3\right){\left(a+1\right)}^{2}& & \\ & =\left[a\left(a–1\right)+3\left(a–1\right)\right]{\left(a+1\right)}^{2}& & \\ & =\left(a+3\right)\left(a–1\right){\left(a+1\right)}^{2}& & \\ & =\left(a+3\right)\left(a–1\right)\left(a+1\right)\left(a+1\right)& & \end{array}$
26. $\begin{array}{cc}{a}^{2}{x}^{3}+2a{x}^{3}–8{a}^{2}–16a& =a{x}^{3}\left(a+2\right)–8a\left(a+2\right)\\ & =\left(a+2\right)\left(a{x}^{3}–8a\right)\\ & =a\left(a+2\right)\left({x}^{3}–8\right)\\ & =a\left(a+2\right)\left(x–2\right)\left({x}^{2}+2x+4\right)\end{array}$
27. $\begin{array}{cc}1–{a}^{6}{b}^{6}& =\left(1–{a}^{3}{b}^{3}\right)\left(1+{a}^{3}{b}^{3}\right)\\ & =\left(1–ab\right)\left(1+ab+{a}^{2}{b}^{2}\right)\left(1+ab\right)\left(1–ab+{a}^{2}{b}^{2}\right)\end{array}$
28. $\begin{array}{cc}5a{x}^{3}+10a{x}^{2}–5ax–10a& =5a{x}^{3}–5ax+10a{x}^{2}–10a\\ & =5ax\left({x}^{2}–1\right)+10a\left({x}^{2}–1\right)\\ & =\left({x}^{2}–1\right)\left(5ax+10a\right)\\ & =5a\left(x–1\right)\left(x+1\right)\left(x+2\right)\end{array}$
29. $\begin{array}{cc}{a}^{2}{x}^{2}+{b}^{2}{y}^{2}–{b}^{2}{x}^{2}–{a}^{2}{y}^{2}& ={a}^{2}{x}^{2}–{a}^{2}{y}^{2}–{b}^{2}{x}^{2}+{b}^{2}{y}^{2}\\ & ={a}^{2}\left({x}^{2}–{y}^{2}\right)–{b}^{2}\left({x}^{2}–{y}^{2}\right)\\ & =\left({x}^{2}–{y}^{2}\right)\left({a}^{2}–{b}^{2}\right)\\ & =\left(x+y\right)\left(x–y\right)\left(a+b\right)\left(a–b\right)\end{array}$
30. $\begin{array}{cccc}{x}^{8}+{x}^{4}–2& ={x}^{8}–{x}^{4}+2{x}^{4}–2& & .\begin{array}{c}2\\ 1\end{array}|\begin{array}{c}2\\ \end{array}\\ & ={x}^{4}\left({x}^{4}–1\right)+2\left({x}^{4}–1\right)& & \\ & =\left({x}^{4}–1\right)\left({x}^{4}+2\right)& & \\ & =\left({x}^{2}–1\right)\left({x}^{2}+1\right)\left({x}^{4}+2\right)& & \\ & =\left(x–1\right)\left(x+1\right)\left({x}^{2}+1\right)\left({x}^{4}+2\right)& & \end{array}$
31. $\begin{array}{cc}{a}^{4}+{a}^{3}–9{a}^{2}–9a& ={a}^{3}\left(a+1\right)–9a\left(a+1\right)\\ & =\left(a+1\right)\left({a}^{3}–9a\right)\\ & =a\left(a+1\right)\left({a}^{2}–9\right)\\ & =a\left(a+1\right)\left(a–3\right)\left(a+3\right)\end{array}$
32. $\begin{array}{cc}{a}^{2}{x}^{2}+{a}^{2}x–6{a}^{2}–{x}^{2}–x+6& ={a}^{2}\left({x}^{2}+x–6\right)–\left({x}^{2}+x–6\right)\\ & =\left({x}^{2}+x–6\right)\left({a}^{2}–1\right)\\ & =\left({x}^{2}–2x+3x–6\right)\left({a}^{2}–1\right)\\ & =\left[x\left(x–2\right)+3\left(x–2\right)\right]\left({a}^{2}–1\right)\\ & =\left(x+3\right)\left(x–2\right)\left(a+1\right)\left(a–1\right)\end{array}$
33. $\begin{array}{cc}16{m}^{4}–25{m}^{2}+9& =16{m}^{4}–25{m}^{2}+9+{m}^{2}–{m}^{2}\\ & =16{m}^{4}–24{m}^{2}+9–{m}^{2}\\ & ={\left(4{m}^{2}–3\right)}^{2}–{m}^{2}\\ & =\left[\left(4{m}^{2}–3\right)–m\right]\left[\left(4{m}^{2}–3\right)+m\right]\\ & =\left(4{m}^{2}–m–3\right)\left(4{m}^{2}+m–3\right)\\ & =\left(4{m}^{2}–4m+3m–3\right)\left(4{m}^{2}+4m–3m–3\right)\\ & =\left[4m\left(m–1\right)+3\left(m–1\right)\right]\left[4m\left(m+1\right)–3\left(m+1\right)\right]\\ & =\left(m–1\right)\left(4m+3\right)\left(m+1\right)\left(4m–3\right)\end{array}$
34. $\begin{array}{cc}3ab{x}^{2}–12ab+3b{x}^{2}–12b& =3ab{x}^{2}+3b{x}^{2}–12ab–12b\\ & =3b{x}^{2}\left(a+1\right)–12b\left(a+1\right)\\ & =\left(a+1\right)\left(3b{x}^{2}–12b\right)\\ & =3b\left(a+1\right)\left({x}^{2}–4\right)\\ & =3b\left(a+1\right)\left(x–2\right)\left(x+2\right)\end{array}$
35. $\begin{array}{cccc}3{a}^{2}m+9am–30m+3{a}^{2}+9a–30& =3m\left({a}^{2}+3a–10\right)+3\left({a}^{2}+3a–10\right)& & .\begin{array}{c}10\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 5\\ \end{array}\\ & =\left({a}^{2}+3a–10\right)\left(3m+3\right)& & \\ & =3\left({a}^{2}–2a+5a–10\right)\left(m+1\right)& & \\ & =3\left[a\left(a–2\right)+5\left(a–2\right)\right]\left(m+1\right)& & \\ & =3\left(a–2\right)\left(a+5\right)\left(m+1\right)& & \end{array}$
36. $\begin{array}{cccc}{a}^{3}{x}^{2}–5{a}^{3}x+6{a}^{3}+{x}^{2}–5x+6& ={a}^{3}\left({x}^{2}–5x+6\right)+\left({x}^{2}–5x+6\right)& & .\begin{array}{c}6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 3\\ \end{array}\\ & =\left({x}^{2}–2x–3x+6\right)\left({a}^{3}+1\right)& & \\ & =\left[x\left(x–2\right)–3\left(x–2\right)\right]\left({a}^{3}+1\right)& & \\ & =\left(x–3\right)\left(x–2\right)\left({a}^{3}+1\right)& & \\ & =\left(x–3\right)\left(x–2\right)\left(a+1\right)\left({a}^{2}–a+1\right)& & \end{array}$
37. $\begin{array}{cc}{x}^{2}\left({x}^{2}–{y}^{2}\right)–\left(2x–1\right)\left({x}^{2}–{y}^{2}\right)& =\left({x}^{2}–{y}^{2}\right)\left[{x}^{2}–\left(2x–1\right)\right]\\ & =\left(x–y\right)\left(x+y\right)\left({x}^{2}–2x+1\right)\\ & =\left(x–y\right)\left(x+y\right){\left(x–1\right)}^{2}\\ & =\left(x–y\right)\left(x+y\right)\left(x–1\right)\left(x–1\right)\end{array}$
38. $\begin{array}{cc}a\left({x}^{3}+1\right)+3ax\left(x+1\right)& =a\left[\left({x}^{3}+1\right)+3x\left(x+1\right)\right]\\ & =a\left[\left(x+1\right)\left({x}^{2}–x+1\right)+3x\left(x+1\right)\right]\\ & =a\left[\left(x+1\right)\left\{\left({x}^{2}–x+1\right)+3x\right\}\right]\\ & =a\left[\left(x+1\right)\left\{{x}^{2}–x+1+3x\right\}\right]\\ & =a\left(x+1\right)\left({x}^{2}+2x+1\right)\\ & =a\left(x+1\right){\left(x+1\right)}^{2}\\ & =a\left(x+1\right)\left(x+1\right)\left(x+1\right)\end{array}$

## Ejercicio 107

CAPITULO X

Descomposición Factorial
Ejercicio 107
Descomponer en tres factores:
1. $\begin{array}{cc}3a{x}^{2}–3a& =3a\left({x}^{2}–1\right)\\ & =3a\left(x–1\right)\left(x+1\right)\end{array}$
2. $\begin{array}{cc}3{x}^{2}–3x–6& =3\left({x}^{2}–x–2\right)\\ & =3\left({x}^{2}+x–2x–2\right)\\ & =3\left[x\left(x+1\right)–2\left(x+1\right)\right]\\ & =3\left(x+1\right)\left(x–2\right)\end{array}$
3. $\begin{array}{cc}2{a}^{2}x–4abx+2{b}^{2}x& =2x\left({a}^{2}–2ab+{b}^{2}\right)\\ & =2x{\left(a–b\right)}^{2}\end{array}$
4. $\begin{array}{cc}2{a}^{3}–2& =2\left({a}^{3}–1\right)\\ & =2\left(a–1\right)\left({a}^{2}+a+1\right)\end{array}$
5. $\begin{array}{cccc}{a}^{3}–3{a}^{2}–28a& =a\left({a}^{2}–3a–28\right)& & .\begin{array}{c}28\\ 14\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 7\\ \end{array}\\ & =a\left({a}^{2}–4a+7a–28\right)& & \\ & =a\left[a\left(a–4\right)+7\left(a–4\right)\right]& & \\ & =a\left(a–4\right)\left(a+7\right)& & \end{array}$
6. $\begin{array}{cc}{x}^{3}–4x+{x}^{2}–4& =x\left({x}^{2}–4\right)+\left({x}^{2}–4\right)\\ & =\left({x}^{2}–4\right)\left(x+1\right)\\ & =\left(x–2\right)\left(x+2\right)\left(x+1\right)\end{array}$
7. $\begin{array}{cc}3a{x}^{3}+3a{y}^{3}& =3a\left({x}^{3}+{y}^{3}\right)\\ & =3a\left(x+y\right)\left({x}^{2}–xy+{y}^{2}\right)\end{array}$
8. $\begin{array}{cc}4a{b}^{2}–4abn+a{n}^{2}& =a\left(4{b}^{2}–4bn+{n}^{2}\right)\\ & =a{\left(2b–n\right)}^{2}\end{array}$
9. $\begin{array}{cccc}{x}^{4}–3{x}^{2}–4& ={x}^{4}+{x}^{2}–4{x}^{2}–4& & .\begin{array}{c}4\\ 1\end{array}|\begin{array}{c}4\\ \end{array}\\ & ={x}^{2}\left({x}^{2}+1\right)–4\left({x}^{2}+1\right)& & \\ & =\left({x}^{2}+1\right)\left({x}^{2}–4\right)& & \\ & =\left({x}^{2}+1\right)\left(x–2\right)\left(x+2\right)& & \end{array}$
10. $\begin{array}{cc}{a}^{3}–{a}^{2}–a+1& ={a}^{2}\left(a–1\right)–\left(a–1\right)\\ & =\left(a–1\right)\left({a}^{2}–1\right)\\ & =\left(a–1\right)\left(a–1\right)\left(a+1\right)\\ & ={\left(a–1\right)}^{2}\left(a+1\right)\end{array}$
11. $\begin{array}{cc}2a{x}^{2}–4ax+2a& =2a\left({x}^{2}–2x+1\right)\\ & =2a{\left(x–1\right)}^{2}\end{array}$
12. $\begin{array}{cc}{x}^{3}–x+{x}^{2}y–y& =x\left({x}^{2}–1\right)+y\left({x}^{2}–1\right)\\ & =\left({x}^{2}–1\right)\left(x+y\right)\\ & =\left(x+1\right)\left(x–1\right)\left(x+y\right)\end{array}$
13. $\begin{array}{cccc}2{a}^{3}+6{a}^{2}–8a& =2a\left({a}^{2}+3a–4\right)& & .\begin{array}{c}4\\ 1\end{array}|\begin{array}{c}4\\ \end{array}\\ & =2a\left({a}^{2}–a+4a–4\right)& & \\ & =2a\left[a\left(a–1\right)+4\left(a–1\right)\right]& & \\ & =2a\left(a–1\right)\left(a+4\right)& & \end{array}$
14. $\begin{array}{cc}16{x}^{3}–48{x}^{2}y+36x{y}^{2}& =4x\left(4{x}^{2}–12xy+9{y}^{2}\right)\\ & =4x{\left(2x–3y\right)}^{2}\end{array}$
15. $\begin{array}{cc}3{x}^{3}–{x}^{2}y–3x{y}^{2}+{y}^{3}& =3{x}^{3}–3x{y}^{2}–{x}^{2}y+{y}^{3}\\ & =3x\left({x}^{2}–{y}^{2}\right)–y\left({x}^{2}–{y}^{2}\right)\\ & =\left({x}^{2}–{y}^{2}\right)\left(3x–y\right)\\ & =\left(x+y\right)\left(x–y\right)\left(3x–y\right)\end{array}$
16. $\begin{array}{cc}5{a}^{4}+5a& =5a\left({a}^{3}+1\right)\\ & =5a\left(a+1\right)\left({a}^{2}–a+1\right)\end{array}$
17. $\begin{array}{cccc}6a{x}^{2}–ax–2a& =a\left(6{x}^{2}–x–2\right)& & .\begin{array}{c}12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ \end{array}\\ & =a\left(6{x}^{2}+3x–4x–2\right)& & \\ & =a\left[3x\left(2x+1\right)–2\left(2x+1\right)\right]& & \\ & =a\left(2x+1\right)\left(3x–2\right)& & \end{array}$
18. $\begin{array}{cc}{n}^{4}–81& =\left({n}^{2}+9\right)\left({n}^{2}–9\right)\\ & =\left({n}^{2}+9\right)\left(n+3\right)\left(n–3\right)\end{array}$
19. $\begin{array}{cc}8a{x}^{2}–2a& =2a\left(4{x}^{2}–1\right)\\ & =2a\left(2x–1\right)\left(2x+1\right)\end{array}$
20. $\begin{array}{cc}a{x}^{3}+10a{x}^{2}+25ax& =ax\left({x}^{2}+10x+25\right)\\ & =ax{\left(x+5\right)}^{2}\end{array}$
21. $\begin{array}{cccc}{x}^{3}–6{x}^{2}–7x& =x\left({x}^{2}–6x–7\right)& & .\begin{array}{c}7\\ 1\end{array}|\begin{array}{c}7\\ \end{array}\\ & =x\left({x}^{2}+x–7x–7\right)& & \\ & =x\left[x\left(x+1\right)–7\left(x+1\right)\right]& & \\ & =x\left(x+1\right)\left(x–7\right)& & \end{array}$
22. $\begin{array}{cc}{m}^{3}+3{m}^{2}–16m–48& ={m}^{2}\left(m+3\right)–16\left(m+3\right)\\ & =\left(m+3\right)\left({m}^{2}–16\right)\\ & =\left(m+3\right)\left(m–4\right)\left(m+4\right)\end{array}$
23. $\begin{array}{cc}{x}^{3}–6{x}^{2}y+12x{y}^{2}–8{y}^{3}& ={\left(x–2y\right)}^{3}\\ & =\left(x–2y\right)\left(x–2y\right)\left(x–2y\right)\end{array}$
24. $\begin{array}{cc}\left(a+b\right)\left({a}^{2}–{b}^{2}\right)–\left({a}^{2}–{b}^{2}\right)& =\left({a}^{2}–{b}^{2}\right)\left[\left(a+b\right)–1\right]\\ & =\left({a}^{2}–{b}^{2}\right)\left(a+b–1\right)\\ & =\left(a–b\right)\left(a+b\right)\left(a+b–1\right)\end{array}$
25. $\begin{array}{cc}32{a}^{5}x–48{a}^{3}bx+18a{b}^{2}x& =2ax\left(16{a}^{4}–24{a}^{2}b+9{b}^{2}\right)\\ & =2ax{\left(4{a}^{2}–3b\right)}^{2}\end{array}$
26. $\begin{array}{cc}{x}^{4}–{x}^{3}+{x}^{2}–x& ={x}^{3}\left(x–1\right)+x\left(x–1\right)\\ & =\left(x–1\right)\left({x}^{3}+x\right)\\ & =x\left(x–1\right)\left({x}^{2}+1\right)\end{array}$
27. $\begin{array}{cccc}4{x}^{2}+32x–36& =4\left({x}^{2}+8x–9\right)& & .\begin{array}{c}9\\ 1\end{array}|\begin{array}{c}9\\ \end{array}\\ & =4\left({x}^{2}–x+9x–9\right)& & \\ & =4\left[x\left(x–1\right)+9\left(x–1\right)\right]& & \\ & =4\left(x–1\right)\left(x+9\right)& & \end{array}$
28. $\begin{array}{cc}{a}^{4}–{\left(a+2\right)}^{2}& =\left[{a}^{2}–\left(a+2\right)\right]\left[{a}^{2}+\left(a+2\right)\right]\\ & =\left({a}^{2}–a–2\right)\left({a}^{2}+a+2\right)\end{array}$
29. $\begin{array}{cccc}{x}^{6}–25{x}^{3}–54& ={x}^{6}–27{x}^{3}+2{x}^{3}–54& & .\begin{array}{c}54\\ 27\\ 1\end{array}|\begin{array}{c}2\\ 27\\ \end{array}\\ & ={x}^{3}\left({x}^{3}–27\right)+2\left({x}^{3}–27\right)& & \\ & =\left({x}^{3}–27\right)\left({x}^{3}+2\right)& & \\ & =\left(x–3\right)\left({x}^{2}+3x+9\right)\left({x}^{3}+2\right)& & \end{array}$
30. $\begin{array}{cc}{a}^{6}+a& =a\left({a}^{5}+1\right)\\ & =a\left(a+1\right)\left({a}^{4}–{a}^{3}+{a}^{2}–a+1\right)\end{array}$
31. $\begin{array}{cc}{a}^{3}b+2{a}^{2}bx+ab{x}^{2}–ab{y}^{2}& =ab\left({a}^{2}+2ax+{x}^{2}–{y}^{2}\right)\\ & =ab\left[{\left(a+x\right)}^{2}–{y}^{2}\right]\\ & =ab\left[\left(a+x\right)+y\right]\left[\left(a+x\right)–y\right]\\ & =ab\left(a+x+y\right)\left(a+x–y\right)\end{array}$
32. $\begin{array}{cc}3ab{m}^{2}–3ab& =3ab\left({m}^{2}–1\right)\\ & =3ab\left(m–1\right)\left(m+1\right)\end{array}$
33. $\begin{array}{cc}81{x}^{4}y+3x{y}^{4}& =3xy\left(27{x}^{3}+{y}^{3}\right)\\ & =3xy\left(3x+y\right)\left(9{x}^{2}–3xy+{y}^{2}\right)\end{array}$
34. $\begin{array}{cc}{a}^{4}–{a}^{3}+a–1& ={a}^{3}\left(a–1\right)+\left(a–1\right)\\ & =\left(a–1\right)\left({a}^{3}+1\right)\\ & =\left(a–1\right)\left(a+1\right)\left({a}^{2}–a+1\right)\end{array}$
35. $\begin{array}{cccc}x–3{x}^{2}–18{x}^{3}& =x\left(1–3x–18{x}^{2}\right)& & .\begin{array}{c}18\\ 9\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ \end{array}\\ & =x\left(1–6x+3x–18{x}^{2}\right)& & \\ & =x\left[\left(1–6x\right)+3x\left(1–6x\right)\right]& & \\ & =x\left(1–6x\right)\left(1+3x\right)& & \end{array}$
36. $\begin{array}{cc}6ax–2bx+6ab–2{b}^{2}& =6ax+6ab–2bx–2{b}^{2}\\ & =6a\left(x+b\right)–2b\left(x+b\right)\\ & =\left(x+b\right)\left(6a–2b\right)\\ & =2\left(x+b\right)\left(3a–b\right)\end{array}$
37. $\begin{array}{cccc}a{m}^{3}–7a{m}^{2}+12am& =am\left({m}^{2}–7m+12\right)& & .\begin{array}{c}12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ \end{array}\\ & =am\left({m}^{2}–4m–3m+12\right)& & \\ & =am\left[m\left(m–4\right)–3\left(m–4\right)\right]& & \\ & =am\left(m–4\right)\left(m–3\right)& & \end{array}$
38. $\begin{array}{cc}4{a}^{2}{x}^{3}–4{a}^{2}& =4{a}^{2}\left({x}^{3}–1\right)\\ & =4{a}^{2}\left(x–1\right)\left({x}^{2}+x+1\right)\end{array}$
39. $\begin{array}{cc}28{x}^{3}y–7x{y}^{3}& =7xy\left(4{x}^{2}–{y}^{2}\right)\\ & =7xy\left(2x–y\right)\left(2x+y\right)\end{array}$
40. $\begin{array}{cccc}3ab{x}^{2}–3abx–18ab& =3ab\left({x}^{2}–x–6\right)& & .\begin{array}{c}6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 3\\ \end{array}\\ & =3ab\left({x}^{2}–3x+2x–6\right)& & \\ & =3ab\left[x\left(x–3\right)+2\left(x–3\right)\right]& & \\ & =3ab\left(x–3\right)\left(x+2\right)& & \end{array}$
41. $\begin{array}{cccc}{x}^{4}–8{x}^{2}–128& ={x}^{4}–16{x}^{2}+8{x}^{2}–128& & .\begin{array}{c}128\\ 64\\ 32\\ 16\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 16\\ \end{array}\\ & ={x}^{2}\left({x}^{2}–16\right)+8\left({x}^{2}–16\right)& & \\ & =\left({x}^{2}–16\right)\left({x}^{2}+8\right)& & \\ & =\left(x+4\right)\left(x–4\right)\left({x}^{2}+8\right)& & \end{array}$
42. $\begin{array}{cc}18{x}^{2}y+60x{y}^{2}+50{y}^{3}& =2y\left(9{x}^{2}+30xy+25{y}^{2}\right)\\ & =2y{\left(3x+5y\right)}^{2}\end{array}$
43. $\begin{array}{cc}\left({x}^{2}–2xy\right)\left(a+1\right)+{y}^{2}\left(a+1\right)& =\left(a+1\right)\left[\left({x}^{2}–2xy\right)+{y}^{2}\right]\\ & =\left(a+1\right)\left[{x}^{2}–2xy+{y}^{2}\right]\\ & =\left(a+1\right){\left(x–y\right)}^{2}\end{array}$
44. $\begin{array}{cccc}{x}^{3}+2{x}^{2}y–3x{y}^{2}& =x\left({x}^{2}+2xy–3{y}^{2}\right)& & .\begin{array}{c}3\\ 1\end{array}|\begin{array}{c}3\\ \end{array}\\ & =x\left({x}^{2}–xy+3xy–3{y}^{2}\right)& & \\ & =x\left[x\left(x–y\right)+3y\left(x–y\right)\right]& & \\ & =x\left(x–y\right)\left(x+3y\right)& & \end{array}$
45. $\begin{array}{cc}{a}^{2}x–4{b}^{2}x+2{a}^{2}y–8{b}^{2}y& ={a}^{2}x+2{a}^{2}y–4{b}^{2}x–8{b}^{2}y\\ & ={a}^{2}\left(x+2y\right)–4{b}^{2}\left(x+2y\right)\\ & ={a}^{2}\left(x+2y\right)\left({a}^{2}–4{b}^{2}\right)\\ & ={a}^{2}\left(x+2y\right)\left(a–2b\right)\left(a+2b\right)\end{array}$
46. $\begin{array}{cc}45{a}^{2}{x}^{4}–20{a}^{2}& =5{a}^{2}\left(9{x}^{4}–4\right)\\ & =5{a}^{2}\left(3{x}^{2}–2\right)\left(3{x}^{2}+2\right)\end{array}$
47. $\begin{array}{cc}{a}^{4}–{\left(a–12\right)}^{2}& =\left[{a}^{2}–\left(a–12\right)\right]\left[{a}^{2}+\left(a–12\right)\right]\\ & =\left({a}^{2}–a+12\right)\left({a}^{2}+a–12\right)\end{array}$
48. $\begin{array}{cc}b{x}^{2}–b–{x}^{2}+1& =b\left({x}^{2}–1\right)–\left({x}^{2}–1\right)\\ & =\left({x}^{2}–1\right)\left(b–1\right)\\ & =\left(x–1\right)\left(x+1\right)\left(b–1\right)\end{array}$
49. $\begin{array}{cccc}2{x}^{4}+6{x}^{3}–56{x}^{2}& =2{x}^{2}\left({x}^{2}+3x–28\right)& & .\begin{array}{c}28\\ 14\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 7\\ \end{array}\\ & =2{x}^{2}\left({x}^{2}–4x+7x–28\right)& & \\ & =2{x}^{2}\left[x\left(x–4\right)+7\left(x–4\right)\right]& & \\ & =2{x}^{2}\left(x–4\right)\left(x+7\right)& & \end{array}$
50. $\begin{array}{cccc}30{a}^{2}–55a–50& =5\left(6{a}^{2}–11a–10\right)& & .\begin{array}{c}60\\ 30\\ 15\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 15\\ \end{array}\\ & =5\left(6{a}^{2}–15a+4a–10\right)& & \\ & =5\left[3a\left(2a–5\right)+2\left(2a–5\right)\right]& & \\ & =5\left(2a–5\right)\left(3a+2\right)& & \end{array}$
51. $\begin{array}{cc}9{\left(x–y\right)}^{3}–\left(x–y\right)& =\left(x–y\right)\left[9{\left(x–y\right)}^{2}–1\right]\\ & =\left(x–y\right)\left[3\left(x–y\right)–1\right]\left[3\left(x–y\right)+1\right]\\ & =\left(x–y\right)\left(3x–3y–1\right)\left(3x–3y+1\right)\end{array}$
52. $\begin{array}{cc}6{a}^{2}x–9{a}^{3}–a{x}^{2}& =–a{x}^{2}+6{a}^{2}x–9{a}^{3}\\ & =–a\left({x}^{2}–6ax+9{a}^{2}\right)\\ & =–a{\left(x–3a\right)}^{2}\\ & =–a\left(x–3a\right)\left(x–3a\right)\\ & =a\left(3a–x\right)\left(x–3a\right)\end{array}$
53. $\begin{array}{cc}64a–125{a}^{4}& =a\left(64–125{a}^{3}\right)\\ & =a\left(4–5a\right)\left(16+20a+25{a}^{2}\right)\end{array}$
54. $\begin{array}{cccc}70{x}^{4}+26{x}^{3}–24{x}^{2}& =2{x}^{2}\left(35{x}^{2}+13x–12\right)& & .\begin{array}{c}420\\ 210\\ 105\\ 35\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 5\\ 7\\ \end{array}\\ & =2{x}^{2}\left(35{x}^{2}–15x+28x–12\right)& & \\ & =2{x}^{2}\left[5x\left(7x–3\right)+4\left(7x–3\right)\right]& & \\ & =2{x}^{2}\left(7x–3\right)\left(5x+4\right)& & \end{array}$
55. $\begin{array}{cccc}{a}^{7}+6{a}^{5}–55{a}^{3}& ={a}^{3}\left({a}^{4}+6{a}^{2}–55\right)& & .\begin{array}{c}55\\ 11\\ 1\end{array}|\begin{array}{c}5\\ 11\\ \end{array}\\ & ={a}^{3}\left({a}^{4}+11{a}^{2}–5{a}^{2}–55\right)& & \\ & ={a}^{3}\left[{a}^{2}\left({a}^{2}+11\right)–5\left({a}^{2}+11\right)\right]& & \\ & ={a}^{3}\left({a}^{2}+11\right)\left({a}^{2}–5\right)& & \end{array}$
56. $\begin{array}{cc}16{a}^{5}b–56{a}^{3}{b}^{3}+49a{b}^{5}& =ab\left(16{a}^{4}–56{a}^{2}{b}^{2}+49{b}^{4}\right)\\ & =ab{\left(4{a}^{2}–7{b}^{2}\right)}^{2}\end{array}$
57. $\begin{array}{cccc}7{x}^{6}+32{a}^{2}{x}^{4}–15{a}^{4}{x}^{2}& ={x}^{2}\left(7{x}^{4}+32{a}^{2}{x}^{2}–15{a}^{4}\right)& & .\begin{array}{c}105\\ 35\\ 1\end{array}|\begin{array}{c}3\\ 35\\ \end{array}\\ & ={x}^{2}\left(7{x}^{4}–3{a}^{2}{x}^{2}+35{a}^{2}{x}^{2}–15{a}^{4}\right)& & \\ & ={x}^{2}\left[{x}^{2}\left(7{x}^{2}–3{a}^{2}\right)+5{a}^{2}\left(7{x}^{2}–3{a}^{2}\right)\right]& & \\ & ={x}^{2}\left(7{x}^{2}–3{a}^{2}\right)\left({x}^{2}+5{a}^{2}\right)& & \end{array}$
58. $\begin{array}{cc}{x}^{2m+2}–{x}^{2}{y}^{2n}& ={x}^{2}\left({x}^{2m}–{y}^{2n}\right)\\ & ={x}^{2}\left({x}^{m}–{y}^{n}\right)\left({x}^{m}+{y}^{n}\right)\end{array}$
59. $\begin{array}{cc}2{x}^{4}+5{x}^{3}–54x–135& =2{x}^{4}–54x+5{x}^{3}–135\\ & =2x\left({x}^{3}–27\right)+5\left({x}^{3}–27\right)\\ & =\left({x}^{3}–27\right)\left(2x+5\right)\\ & =\left(x–3\right)\left({x}^{2}+3x+9\right)\left(2x+5\right)\end{array}$
60. $\begin{array}{cc}a{x}^{3}+a{x}^{2}y+ax{y}^{2}–2a{x}^{2}–2axy–2a{y}^{2}& =ax\left({x}^{2}+xy+{y}^{2}\right)–2a\left({x}^{2}+xy+{y}^{2}\right)\\ & =\left({x}^{2}+xy+{y}^{2}\right)\left(ax–2a\right)\\ & =a\left({x}^{2}+xy+{y}^{2}\right)\left(x–2\right)\end{array}$
61. $\begin{array}{cc}{\left(x+y\right)}^{4}–1& =\left[{\left(x+y\right)}^{2}–1\right]\left[{\left(x+y\right)}^{2}+1\right]\\ & =\left[\left(x+y\right)–1\right]\left[\left(x+y\right)+1\right]\left[{x}^{2}+2xy+{y}^{2}+1\right]\\ & =\left(x+y–1\right)\left(x+y+1\right)\left({x}^{2}+2xy+{y}^{2}+1\right)\end{array}$
62. $\begin{array}{cc}3{a}^{5}+3{a}^{3}+3a& =3a\left({a}^{4}+{a}^{2}+1\right)\\ & =3a\left({a}^{4}+{a}^{2}+1+{a}^{2}–{a}^{2}\right)\\ & =3a\left({a}^{4}+2{a}^{2}+1–{a}^{2}\right)\\ & =3a\left\{{\left({a}^{2}+1\right)}^{2}–{a}^{2}\right\}\\ & =3a\left\{\left[\left({a}^{2}+1\right)–a\right]\left[\left({a}^{2}+1\right)+a\right]\right\}\\ & =3a\left({a}^{2}–a+1\right)\left({a}^{2}+a+1\right)\end{array}$

## Ejercicio 106

CAPITULO X

Descomposición Factorial
Ejercicio 106
MISCELANIA SOBRE LOS 10 CASOS DE DESCOMPOSICION EN FACTORES
Descomponer en factores:
1. $5{a}^{2}+a=a\left(5a+1\right)$
2. ${m}^{2}+2mx+{x}^{2}={\left(m+x\right)}^{2}$
3. $\begin{array}{cc}{a}^{2}+a–ab–b& =a\left(a+1\right)–b\left(a+1\right)\\ & =\left(a+1\right)\left(a–b\right)\end{array}$
4. ${x}^{2}–36=\left(x–6\right)\left(x+6\right)$
5. $9{x}^{2}–6xy+{y}^{2}={\left(3x–y\right)}^{2}$
6. $\begin{array}{cccc}{x}^{2}–3x–4& ={x}^{2}+x–4x–4& & .\begin{array}{c}4\\ 1\end{array}|\begin{array}{c}4\\ \end{array}\\ & =x\left(x+1\right)–4\left(x+1\right)& & \\ & =\left(x+1\right)\left(x–4\right)& & \end{array}$
7. $\begin{array}{cccc}6{x}^{2}–x–2& =6{x}^{2}+3x–4x–2& & .\begin{array}{c}12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ \end{array}\\ & =3x\left(2x+1\right)–2\left(2x+1\right)& & \\ & =\left(2x+1\right)\left(3x–2\right)& & \end{array}$
8. $1+{x}^{3}=\left(1+x\right)\left(1–x+{x}^{2}\right)$
9. $\begin{array}{cc}27{a}^{3}–1& =\left(3a–1\right)\left[{\left(3a\right)}^{2}+3a+1\right]\\ & =\left(3a–1\right)\left(9{a}^{2}+3a+1\right)\end{array}$
10. ${x}^{5}+{m}^{5}=\left(x+m\right)\left({x}^{4}–{x}^{3}m+{x}^{2}{m}^{2}–x{m}^{3}+{m}^{4}\right)$
11. ${a}^{3}–3{a}^{2}b+5a{b}^{2}=a\left({a}^{2}–3ab+5{b}^{2}\right)$
12. $\begin{array}{cc}2xy–6y+xz–3z& =2y\left(x–3\right)+z\left(x–3\right)\\ & =\left(x–3\right)\left(2y+z\right)\end{array}$
13. $1–4b+4{b}^{2}={\left(1–2b\right)}^{2}$
14. $\begin{array}{cc}4{x}^{4}+3{x}^{2}{y}^{2}+{y}^{4}& =4{x}^{4}+3{x}^{2}{y}^{2}+{y}^{4}+{x}^{2}{y}^{2}–{x}^{2}{y}^{2}\\ & =4{x}^{4}+4{x}^{2}{y}^{2}+{y}^{4}–{x}^{2}{y}^{2}\\ & ={\left(2{x}^{2}+{y}^{2}\right)}^{2}–{x}^{2}{y}^{2}\\ & =\left[\left(2{x}^{2}+{y}^{2}\right)+xy\right]\left[\left(2{x}^{2}+{y}^{2}\right)–xy\right]\\ & =\left(2{x}^{2}+xy+{y}^{2}\right)\left(2{x}^{2}–xy+{y}^{2}\right)\end{array}$
15. $\begin{array}{cc}{x}^{8}–6{x}^{4}{y}^{4}+{y}^{8}& ={x}^{8}–6{x}^{4}{y}^{4}+{y}^{8}+4{x}^{4}{y}^{4}–4{x}^{4}{y}^{4}\\ & ={x}^{8}–2{x}^{4}{y}^{4}+{y}^{8}–4{x}^{4}{y}^{4}\\ & ={\left({x}^{4}–{y}^{4}\right)}^{2}–4{x}^{4}{y}^{4}\\ & =\left[\left({x}^{4}–{y}^{4}\right)–2{x}^{2}{y}^{2}\right]\left[\left({x}^{4}–{y}^{4}\right)+2{x}^{2}{y}^{2}\right]\\ & =\left({x}^{4}–2{x}^{2}{y}^{2}–{y}^{4}\right)\left({x}^{4}+2{x}^{2}{y}^{2}–{y}^{4}\right)\end{array}$
16. $\begin{array}{cccc}{a}^{2}–a–30& ={a}^{2}–6a+5a–30& & .\begin{array}{c}30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ \end{array}\\ & =a\left(a–6\right)+5\left(a–6\right)& & \\ & =\left(a–6\right)\left(a+5\right)& & \end{array}$
17. $\begin{array}{cccc}15{m}^{2}+11m–14& =15{m}^{2}–10m+21m–14& & .\begin{array}{c}210\\ 105\\ 35\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ 7\\ \end{array}\\ & =5m\left(3m–2\right)+7\left(3m–2\right)& & \\ & =\left(3m–2\right)\left(5m+7\right)& & \end{array}$
18. $\begin{array}{cc}{a}^{6}+1& =\left({a}^{2}+1\right)\left[{\left({a}^{2}\right)}^{2}–{a}^{2}+1\right]\\ & =\left({a}^{2}+1\right)\left({a}^{4}–{a}^{2}+1\right)\end{array}$
19. $\begin{array}{cc}8{m}^{3}–27{y}^{6}& =\left(2m–3{y}^{2}\right)\left[{\left(2m\right)}^{2}+\left(2m\right)\left(3{y}^{2}\right)+{\left(3{y}^{2}\right)}^{2}\right]\\ & =\left(2m–3{y}^{2}\right)\left(4{m}^{2}+6m{y}^{2}+9{y}^{4}\right)\end{array}$
20. $16{a}^{2}–24ab+9{b}^{2}={\left(4a–3b\right)}^{2}$
21. $1+{a}^{7}=\left(1+a\right)\left(1–a+{a}^{2}–{a}^{3}+{a}^{4}–{a}^{5}+{a}^{6}\right)$
22. $\begin{array}{cc}8{a}^{3}–12{a}^{2}+6a–1& =\left(8{a}^{3}–1\right)–\left(12{a}^{2}–6a\right)\\ & =\left(2a–1\right)\left[{\left(2a\right)}^{2}+2a+1\right]–6a\left(2a–1\right)\\ & =\left(2a–1\right)\left(4{a}^{2}+2a+1\right)–6a\left(2a–1\right)\\ & =\left(2a–1\right)\left[\left(4{a}^{2}+2a+1\right)–6a\right]\\ & =\left(2a–1\right)\left[4{a}^{2}+2a+1–6a\right]\\ & =\left(2a–1\right)\left(4{a}^{2}–4a+1\right)\\ & =\left(2a–1\right){\left(2a–1\right)}^{2}\\ & ={\left(2a–1\right)}^{3}\end{array}$
23. $1–{m}^{2}=\left(1–m\right)\left(1+m\right)$
24. $\begin{array}{cccc}{x}^{4}+4{x}^{2}–21& ={x}^{4}–3{x}^{2}+7{x}^{2}–21& & .\begin{array}{c}21\\ 7\\ 1\end{array}|\begin{array}{c}3\\ 7\\ \end{array}\\ & ={x}^{2}\left({x}^{2}–3\right)+7\left({x}^{2}–3\right)& & \\ & =\left({x}^{2}–3\right)\left({x}^{2}+7\right)& & \end{array}$
25. $\begin{array}{cc}125{a}^{6}+1& =\left(5{a}^{2}+1\right)\left[{\left(5{a}^{2}\right)}^{2}+5{a}^{2}+1\right]\\ & =\left(5{a}^{2}+1\right)\left(25{a}^{4}+5{a}^{2}+1\right)\end{array}$
26. $\begin{array}{cc}{a}^{2}+2ab+{b}^{2}–{m}^{2}& ={\left(a+b\right)}^{2}–{m}^{2}\\ & =\left[\left(a+b\right)–m\right]\left[\left(a+b\right)+m\right]\\ & =\left(a+b–m\right)\left(a+b+m\right)\end{array}$
27. $8{a}^{2}b+16{a}^{3}b–24{a}^{2}{b}^{2}=8{a}^{2}b\left(1+2a–3b\right)$
28. $\begin{array}{cc}{x}^{5}–{x}^{4}+x–1& ={x}^{4}\left(x–1\right)+\left(x–1\right)\\ & =\left(x–1\right)\left({x}^{4}+1\right)\end{array}$
29. $\begin{array}{cccc}6{x}^{2}+19x–20& =6{x}^{2}+24x–5x–20& & .\begin{array}{c}120\\ 60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ 5\\ \end{array}\\ & =6x\left(x+4\right)–5\left(x+4\right)& & \\ & =\left(x+4\right)\left(6x–5\right)& & \end{array}$
30. $25{x}^{4}–81{y}^{2}=\left(5{x}^{2}+9y\right)\left(5{x}^{2}–9y\right)$
31. $1–{m}^{3}=\left(1–m\right)\left(1+m+{m}^{2}\right)$
32. $\begin{array}{cc}{x}^{2}–{a}^{2}+2xy+{y}^{2}+2ab–{b}^{2}& =\left({x}^{2}+2xy+{y}^{2}\right)–\left({a}^{2}–2ab+{b}^{2}\right)\\ & ={\left(x+y\right)}^{2}–{\left(a–b\right)}^{2}\\ & =\left[\left(x+y\right)+\left(a–b\right)\right]\left[\left(x+y\right)–\left(a–b\right)\right]\\ & =\left(x+y+a–b\right)\left(x+y–a+b\right)\end{array}$
33. $21{m}^{5}n–7{m}^{4}{n}^{2}+7{m}^{3}{n}^{3}–7{m}^{2}n=7{m}^{2}n\left(3{m}^{3}–{m}^{2}n+m{n}^{2}–1\right)$
34. $a\left(x+1\right)–b\left(x+1\right)+c\left(x+1\right)=\left(x+1\right)\left(a–b+c\right)$
35. $\begin{array}{cc}4+4\left(x–y\right)+{\left(x–y\right)}^{2}& ={\left[2+\left(x–y\right)\right]}^{2}\\ & ={\left(x–y+2\right)}^{2}\end{array}$
36. $1–{a}^{2}{b}^{4}=\left(1+a{b}^{2}\right)\left(1–a{b}^{2}\right)$
37. ${b}^{2}+12ab+36{a}^{2}={\left(b+6a\right)}^{2}$
38. $\begin{array}{cccc}{x}^{6}+4{x}^{3}–77& ={x}^{6}–7{x}^{3}+11{x}^{3}–77& & .\begin{array}{c}77\\ 11\\ 1\end{array}|\begin{array}{c}7\\ 11\\ \end{array}\\ & ={x}^{3}\left({x}^{3}–7\right)+11\left({x}^{3}–7\right)& & \\ & =\left({x}^{3}–7\right)\left({x}^{3}+11\right)& & \end{array}$
39. $\begin{array}{cccc}15{x}^{4}–17{x}^{2}–4& =15{x}^{4}–20{x}^{2}+3{x}^{2}–4& & .\begin{array}{c}60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 5\\ \end{array}\\ & =5{x}^{2}\left(3{x}^{2}–4\right)+\left(3{x}^{2}–4\right)& & \\ & =\left(3{x}^{2}–4\right)\left(5{x}^{2}+1\right)& & \end{array}$
40. $\begin{array}{cc}1+{\left(a–3b\right)}^{3}& =\left[1+\left(a–3b\right)\right]\left[1–\left(a–3b\right)+{\left(a–3b\right)}^{2}\right]\\ & =\left(a–3b+1\right)\left(1–a+3b+{a}^{2}–6ab+9{b}^{2}\right)\end{array}$
41. $\begin{array}{cc}{x}^{4}+{x}^{2}+25& ={x}^{4}+{x}^{2}+25+9{x}^{2}–9{x}^{2}\\ & ={x}^{4}+10{x}^{2}+25–9{x}^{2}\\ & ={\left({x}^{2}+5\right)}^{2}–9{x}^{2}\\ & =\left[\left({x}^{2}+5\right)–3x\right]\left[\left({x}^{2}+5\right)+3x\right]\\ & =\left({x}^{2}–3x+5\right)\left({x}^{2}+3x+5\right)\end{array}$
42. $\begin{array}{cc}{a}^{8}–28{a}^{4}+36& ={a}^{8}–28{a}^{4}+36+16{a}^{4}–16{a}^{4}\\ & ={a}^{8}–12{a}^{4}+36–16{a}^{4}\\ & ={\left({a}^{4}–6\right)}^{2}–16{a}^{4}\\ & =\left[\left({a}^{4}–6\right)+4{a}^{2}\right]\left[\left({a}^{4}–6\right)–4{a}^{2}\right]\\ & =\left({a}^{4}+4{a}^{2}–6\right)\left({a}^{4}–4{a}^{2}–6\right)\end{array}$
43. $\begin{array}{cc}343+8{a}^{3}& =\left(7+2a\right)\left[{7}^{2}–7\left(2a\right)+{\left(2a\right)}^{2}\right]\\ & =\left(7+2a\right)\left(49–14a+4{a}^{2}\right)\end{array}$
44. $12{a}^{2}bx–15{a}^{2}by=3{a}^{2}b\left(4x–5y\right)$
45. $\begin{array}{cccc}{x}^{2}+2xy–15{y}^{2}& ={x}^{2}–3xy+5xy–15{y}^{2}& & .\begin{array}{c}15\\ 5\\ 1\end{array}|\begin{array}{c}3\\ 5\\ \end{array}\\ & =x\left(x–3y\right)+5y\left(x–3y\right)& & \\ & =\left(x–3y\right)\left(x+5y\right)& & \end{array}$
46. $\begin{array}{cc}6am–4an–2n+3m& =6am+3m–4an–2n\\ & =3m\left(2a+1\right)–2n\left(2a+1\right)\\ & =\left(2a+1\right)\left(3m–2n\right)\end{array}$
47. $81{a}^{6}–4{b}^{2}{c}^{8}=\left(9{a}^{3}–2b{c}^{4}\right)\left(9{a}^{3}+2b{c}^{4}\right)$
48. $\begin{array}{cc}16–{\left(2a+b\right)}^{2}& =\left[4–\left(2a+b\right)\right]\left[4+\left(2a+b\right)\right]\\ & =\left(4–2a+b\right)\left(4+2a+b\right)\end{array}$
49. $\begin{array}{cccc}20–x–{x}^{2}& =20+4x–5x–{x}^{2}& & .\begin{array}{c}20\\ 10\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 5\\ \end{array}\\ & =4\left(5+x\right)–5x\left(5+x\right)& & \\ & =\left(5+x\right)\left(4–5x\right)& & \end{array}$
50. $\begin{array}{cccc}{n}^{2}+n–42& ={n}^{2}–6n+7n–42& & .\begin{array}{c}42\\ 21\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 7\\ \end{array}\\ & =n\left(n–6\right)+7\left(n–6\right)& & \\ & =\left(n–6\right)\left(n+7\right)& & \end{array}$
51. $\begin{array}{cc}{a}^{2}–{d}^{2}+{n}^{2}–{c}^{2}–2an–2cd& =\left({a}^{2}–2an+{n}^{2}\right)–\left({d}^{2}+2cd+{c}^{2}\right)\\ & ={\left(a–n\right)}^{2}–{\left(d+c\right)}^{2}\\ & =\left[\left(a–n\right)+\left(d+c\right)\right]\left[\left(a–n\right)–\left(d+c\right)\right]\\ & =\left(a–n+d+c\right)\left(a–n–d–c\right)\end{array}$
52. $\begin{array}{cc}1+216{x}^{9}& =\left(1+6{x}^{3}\right)\left[1–6{x}^{3}+{\left(6{x}^{3}\right)}^{2}\right]\\ & =\left(1+6{x}^{3}\right)\left(1–6{x}^{3}+36{x}^{6}\right)\end{array}$
53. ${x}^{3}–64=\left(x–4\right)\left({x}^{2}+4x+16\right)$
54. ${x}^{3}–64{x}^{4}={x}^{3}\left(1–64x\right)$
55. $18a{x}^{5}{y}^{3}–36{x}^{4}{y}^{3}–54{x}^{2}{y}^{8}=18{x}^{2}{y}^{3}\left(a{x}^{3}–2{x}^{2}–3{y}^{5}\right)$
56. $49{a}^{2}{b}^{2}–14ab+1={\left(7ab–1\right)}^{2}$
57. $\begin{array}{cc}{\left(x+1\right)}^{2}–81& =\left[\left(x+1\right)–9\right]\left[\left(x+1\right)+9\right]\\ & =\left(x+1–9\right)\left(x+1+9\right)\\ & =\left(x–8\right)\left(x+10\right)\end{array}$
58. $\begin{array}{cc}{a}^{2}–{\left(b+c\right)}^{2}& =\left[a+\left(b+c\right)\right]\left[a–\left(b+c\right)\right]\\ & =\left(a+b+c\right)\left(a–b–c\right)\end{array}$
59. $\begin{array}{cc}{\left(m+n\right)}^{2}–6\left(m+n\right)+9& ={\left[\left(m+n\right)–3\right]}^{2}\\ & ={\left(m+n–3\right)}^{2}\end{array}$
60. $\begin{array}{cccc}7{x}^{2}+31x–20& =7{x}^{2}+35x–4x–20& & .\begin{array}{c}140\\ 70\\ 35\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 35\\ \end{array}\\ & =7x\left(x+5\right)–4\left(x+5\right)& & \\ & =\left(x+5\right)\left(7x–4\right)& & \end{array}$
61. $\begin{array}{cc}9{a}^{3}+63a–45{a}^{2}& =9{a}^{3}–45{a}^{2}+63a\\ & =9a\left({a}^{2}–5a+7\right)\end{array}$
62. $\begin{array}{cc}ax+a–x–1& =a\left(x+1\right)–\left(x+1\right)\\ & =\left(x+1\right)\left(a–1\right)\end{array}$
63. $\begin{array}{cc}81{x}^{4}+25{y}^{2}–90{x}^{2}y& =81{x}^{4}–90{x}^{2}y+25{y}^{2}\\ & ={\left(9{x}^{2}–5y\right)}^{2}\end{array}$
64. $\begin{array}{cc}1–27{b}^{2}+{b}^{4}& =1–27{b}^{2}+{b}^{4}+25{b}^{2}–25{b}^{2}\\ & =1–2{b}^{2}+{b}^{4}–25{b}^{2}\\ & ={\left(1–{b}^{2}\right)}^{2}–25{b}^{2}\\ & =\left[\left(1–{b}^{2}\right)+5b\right]\left[\left(1–{b}^{2}\right)–5b\right]\\ & =\left(1+5b–{b}^{2}\right)\left(1–5b–{b}^{2}\right)\end{array}$
65. $\begin{array}{cc}{m}^{4}+{m}^{2}{n}^{2}+{n}^{4}& ={m}^{4}+{m}^{2}{n}^{2}+{n}^{4}+{m}^{2}{n}^{2}–{m}^{2}{n}^{2}\\ & ={m}^{4}+2{m}^{2}{n}^{2}+{n}^{4}–{m}^{2}{n}^{2}\\ & ={\left({m}^{2}+{n}^{2}\right)}^{2}–{m}^{2}{n}^{2}\\ & =\left[\left({m}^{2}+{n}^{2}\right)–mn\right]\left[\left({m}^{2}+{n}^{2}\right)+mn\right]\\ & =\left({m}^{2}–mn+{n}^{2}\right)\left({m}^{2}+mn+{n}^{2}\right)\end{array}$
66. ${c}^{4}–4{d}^{4}=\left({c}^{2}–2{d}^{2}\right)\left({c}^{2}+2{d}^{2}\right)$
67. $15{x}^{4}–15{x}^{3}+20{x}^{2}=5{x}^{2}\left(3{x}^{2}–3x+4\right)$
68. $\begin{array}{cc}{a}^{2}–{x}^{2}–a–x& =\left(a+x\right)\left(a–x\right)–a–x\\ & =\left(a+x\right)\left(a–x\right)–\left(a+x\right)\\ & =\left(a+x\right)\left[\left(a–x\right)–1\right]\\ & =\left(a+x\right)\left(a–x–1\right)\end{array}$
69. $\begin{array}{cccc}{x}^{4}–8{x}^{2}–240& ={x}^{4}–20{x}^{2}+12{x}^{2}–240& & .\begin{array}{c}240\\ 120\\ 60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 2\\ 3\\ 5\\ \end{array}\\ & ={x}^{2}\left({x}^{2}–20\right)+12\left({x}^{2}–20\right)& & \\ & =\left({x}^{2}–20\right)\left({x}^{2}+12\right)& & \end{array}$
70. $\begin{array}{cccc}6{m}^{4}+7{m}^{2}–20& =6{m}^{4}–8{m}^{2}+15{m}^{2}–20& & .\begin{array}{c}120\\ 60\\ 30\\ 15\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 15\\ \end{array}\\ & =2{m}^{2}\left(3{m}^{2}–4\right)+5\left(3{m}^{2}–4\right)& & \\ & =\left(3{m}^{2}–4\right)\left(2{m}^{2}+5\right)& & \end{array}$
71. $\begin{array}{cc}9{n}^{2}+4{a}^{2}–12an& =9{n}^{2}–12an+4{a}^{2}\\ & ={\left(3n–2a\right)}^{2}\end{array}$
72. $2{x}^{2}+2=2\left({x}^{2}+1\right)$
73. $7a\left(x+y–1\right)–3b\left(x+y–1\right)=\left(x+y–1\right)\left(7a–3b\right)$
74. $\begin{array}{cccc}{x}^{2}+3x–18& ={x}^{2}+6x–3x–18& & .\begin{array}{c}18\\ 9\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ \end{array}\\ & =x\left(x+6\right)–3\left(x+6\right)& & \\ & =\left(x+6\right)\left(x–3\right)& & \end{array}$
75. $\begin{array}{cc}{\left(a+m\right)}^{2}–{\left(b+n\right)}^{2}& =\left[\left(a+m\right)+\left(b+n\right)\right]\left[\left(a+m\right)–\left(b+n\right)\right]\\ & =\left(a+b+m+n\right)\left(a–b+m–n\right)\end{array}$
76. ${x}^{3}+6{x}^{2}y+12x{y}^{2}+8{y}^{3}={\left(x+2y\right)}^{3}$
77. $\begin{array}{cccc}8{a}^{2}–22a–21& =8{a}^{2}+6a–28a–21& & .\begin{array}{c}168\\ 84\\ 42\\ 21\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ 7\\ \end{array}\\ & =2a\left(4a+3\right)–7\left(4a+3\right)& & \\ & =\left(4a+3\right)\left(2a–7\right)& & \end{array}$
78. $1+18ab+81{a}^{2}{b}^{2}={\left(1+9ab\right)}^{2}$
79. $4{a}^{6}–1=\left(2{a}^{3}–1\right)\left(2{a}^{3}+1\right)$
80. $\begin{array}{cccc}{x}^{6}–4{x}^{3}–480& ={x}^{6}–24{x}^{3}+20{x}^{3}–480& & .\begin{array}{c}480\\ 240\\ 120\\ 60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 2\\ 2\\ 3\\ 5\\ \end{array}\\ & ={x}^{3}\left({x}^{3}–24\right)+20\left({x}^{3}–24\right)& & \\ & =\left({x}^{3}–24\right)\left({x}^{3}+20\right)& & \end{array}$
81. $\begin{array}{cc}ax–bx+b–a–by+ay& =ax+ay–a–bx–by+b\\ & =a\left(x+y–1\right)–b\left(x+y–1\right)\\ & =\left(x+y–1\right)\left(a–b\right)\end{array}$
82. $\begin{array}{cc}6am–3m–2a+1& =3m\left(2a–1\right)–\left(2a–1\right)\\ & =\left(2a–1\right)\left(3m–1\right)\end{array}$
83. $\begin{array}{cccc}15+14x–8{x}^{2}& =15+20x–6x–8{x}^{2}& & .\begin{array}{c}120\\ 60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ 5\\ \end{array}\\ & =5\left(3+4x\right)–2x\left(3+4x\right)& & \\ & =\left(3+4x\right)\left(5–2x\right)& & \end{array}$
84. ${a}^{10}–{a}^{8}+{a}^{6}+{a}^{4}={a}^{4}\left({a}^{6}–{a}^{4}+{a}^{2}+1\right)$
85. $\begin{array}{cc}2x\left(a–1\right)–a+1& =2x\left(a–1\right)–\left(a–1\right)\\ & =\left(a–1\right)\left(2x–1\right)\end{array}$
86. $\begin{array}{cc}\left(m+n\right)\left(m–n\right)+3n\left(m–n\right)& =\left(m–n\right)\left[\left(m+n\right)+3n\right]\\ & =\left(m–n\right)\left(m+n+3n\right)\\ & =\left(m–n\right)\left(m+4n\right)\end{array}$
87. $\begin{array}{cc}{a}^{2}–{b}^{3}+2{b}^{3}{x}^{2}–2{a}^{2}{x}^{2}& ={a}^{2}–{b}^{3}–2{a}^{2}{x}^{2}+2{b}^{3}{x}^{2}\\ & ={a}^{2}–{b}^{3}–2{x}^{2}\left({a}^{2}–{b}^{3}\right)\\ & =\left({a}^{2}–{b}^{3}\right)–2{x}^{2}\left({a}^{2}–{b}^{3}\right)\\ & =\left({a}^{2}–{b}^{3}\right)\left(1–2{x}^{2}\right)\end{array}$
88. $\begin{array}{cc}2am–3b–c–cm–3bm+2a& =2am–3bm–cm+2a–3b–c\\ & =m\left(2a–3b–c\right)+2a–3b–c\\ & =m\left(2a–3b–c\right)+\left(2a–3b–c\right)\\ & =\left(2a–3b–c\right)\left(m+1\right)\end{array}$
89. ${x}^{2}–\frac{2}{3}x+\frac{1}{9}={\left(x–\frac{1}{3}\right)}^{2}$
90. $4{a}^{2n}–{b}^{4n}=\left(2{a}^{n}+{b}^{2n}\right)\left(2{a}^{n}–{b}^{2n}\right)$
91. $\begin{array}{cc}81{x}^{2}–{\left(a+x\right)}^{2}& =\left[9x+\left(a+x\right)\right]\left[9x–\left(a+x\right)\right]\\ & =\left[9x+a+x\right]\left[9x–a–x\right]\\ & =\left(10x+a\right)\left(8x–a\right)\end{array}$
92. $\begin{array}{cc}{a}^{2}+9–6a–16{x}^{2}& ={a}^{2}–6a+9–16{x}^{2}\\ & ={\left(a–3\right)}^{2}–16{x}^{2}\\ & =\left[\left(a–3\right)+4x\right]\left[\left(a–3\right)–4x\right]\\ & =\left(a+4x–3\right)\left(a–4x–3\right)\end{array}$
93. $\begin{array}{cc}9{a}^{2}–{x}^{2}–4+4x& =9{a}^{2}–{x}^{2}+4x–4\\ & =9{a}^{2}–\left({x}^{2}–4x+4\right)\\ & =9{a}^{2}–{\left(x–2\right)}^{2}\\ & =\left[3a+\left(x–2\right)\right]\left[3a–\left(x–2\right)\right]\\ & =\left(3a+x–2\right)\left(3a–x+2\right)\end{array}$
94. $\begin{array}{cc}9{x}^{2}–{y}^{2}+3x–y& =\left(3x–y\right)\left(3x+y\right)+\left(3x–y\right)\\ & =\left(3x–y\right)\left[\left(3x+y\right)+1\right]\\ & =\left(3x–y\right)\left(3x+y+1\right)\end{array}$
95. $\begin{array}{cccc}{x}^{2}–x–72& ={x}^{2}–9x+8x–72& & .\begin{array}{c}72\\ 36\\ 18\\ 9\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 9\\ \end{array}\\ & =x\left(x–9\right)+8\left(x–9\right)& & \\ & =\left(x–9\right)\left(x+8\right)& & \end{array}$
96. $\begin{array}{cc}36{a}^{4}–120{a}^{2}{b}^{2}+49{b}^{4}& =36{a}^{4}–120{a}^{2}{b}^{2}+49{b}^{4}+36{a}^{2}{b}^{2}–36{a}^{2}{b}^{2}\\ & =36{a}^{4}–84{a}^{2}{b}^{2}+49{b}^{4}–36{a}^{2}{b}^{2}\\ & ={\left(6{a}^{2}–7{b}^{2}\right)}^{2}–36{a}^{2}{b}^{2}\\ & =\left[\left(6{a}^{2}–7{b}^{2}\right)+6ab\right]\left[\left(6{a}^{2}–7{b}^{2}\right)–6ab\right]\\ & =\left(6{a}^{2}+6ab–7{b}^{2}\right)\left(6{a}^{2}–6ab–7{b}^{2}\right)\end{array}$
97. $\begin{array}{cc}{a}^{2}–{m}^{2}–9{n}^{2}–6mn+4ab+4{b}^{2}& ={a}^{2}+4ab+4{b}^{2}–\left({m}^{2}+6mn+9{n}^{2}\right)\\ & ={\left(a+2b\right)}^{2}–{\left(m+3n\right)}^{2}\\ & =\left[\left(a+2b\right)+\left(m+3n\right)\right]\left[\left(a+2b\right)–\left(m+3n\right)\right]\\ & =\left(a+2b+m+3n\right)\left(a+2b–m–3n\right)\end{array}$
98. $1–\frac{4}{9}{a}^{8}=\left(1–\frac{2}{3}{a}^{4}\right)\left(1+\frac{2}{3}{a}^{4}\right)$
99. $\begin{array}{cc}81{a}^{8}+64{b}^{12}& =81{a}^{8}+64{b}^{12}+144{a}^{4}{b}^{6}–144{a}^{4}{b}^{6}\\ & =81{a}^{8}+144{a}^{4}{b}^{6}+64{b}^{12}–144{a}^{4}{b}^{6}\\ & ={\left(9{a}^{4}+8{b}^{6}\right)}^{2}–144{a}^{4}{b}^{6}\\ & =\left[\left(9{a}^{4}+8{b}^{6}\right)–12{a}^{2}{b}^{3}\right]\left[\left(9{a}^{4}+8{b}^{6}\right)+12{a}^{2}{b}^{3}\right]\\ & =\left(9{a}^{4}–12{a}^{2}{b}^{3}+8{b}^{6}\right)\left(9{a}^{4}+12{a}^{2}{b}^{3}+8{b}^{6}\right)\end{array}$
100. $\begin{array}{cccc}49{x}^{2}–77x+30& =49{x}^{2}–42x–35x+30& & .\begin{array}{c}1470\\ 735\\ 245\\ 49\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ 7\\ 7\\ \end{array}\\ & =7x\left(7x–6\right)–5\left(7x–6\right)& & \\ & =\left(7x–6\right)\left(7x–5\right)& & \end{array}$
101. $\begin{array}{cccc}{x}^{2}–2abx–35{a}^{2}{b}^{2}& ={x}^{2}–7abx+5abx–35{a}^{2}{b}^{2}& & .\begin{array}{c}35\\ 7\\ 1\end{array}|\begin{array}{c}5\\ 7\\ \end{array}\\ & =x\left(x–7ab\right)+5ab\left(x–7ab\right)& & \\ & =\left(x–7ab\right)\left(x+5ab\right)& & \end{array}$
102. $125{x}^{3}–225{x}^{2}+135x–27={\left(5x–3\right)}^{3}$
103. $\begin{array}{cc}{\left(a–2\right)}^{2}–{\left(a+3\right)}^{2}& =\left[\left(a–2\right)+\left(a+3\right)\right]\left[\left(a–2\right)–\left(a+3\right)\right]\\ & =\left[a–2+a+3\right]\left[\overline{)a}–2–\overline{)a}–3\right]\\ & =–5\left(2a+1\right)\end{array}$
104. $\begin{array}{cc}4{a}^{2}m+12{a}^{2}n–5bm–15bn& =4{a}^{2}\left(m+3n\right)–5b\left(m+3n\right)\\ & =\left(m+3n\right)\left(4{a}^{2}–5b\right)\end{array}$
105. $1+6{x}^{3}+9{x}^{6}={\left(1+3{x}^{3}\right)}^{2}$
106. $\begin{array}{cccc}{a}^{4}+3{a}^{2}b–40{b}^{2}& ={a}^{4}+8{a}^{2}b–5{a}^{2}b–40{b}^{2}& & .\begin{array}{c}40\\ 20\\ 10\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 5\\ \end{array}\\ & ={a}^{2}\left({a}^{2}+8b\right)–5b\left({a}^{2}+8b\right)& & \\ & =\left({a}^{2}+8b\right)\left({a}^{2}–5b\right)& & \end{array}$
107. ${m}^{3}+8{a}^{3}{x}^{3}=\left(m+2ax\right)\left({m}^{2}–2amx+4{a}^{2}{x}^{2}\right)$
108. $\begin{array}{cc}1–9{x}^{2}+24xy–16{y}^{2}& =1–\left(9{x}^{2}–24xy+16{y}^{2}\right)\\ & =1–{\left(3x–4y\right)}^{2}\\ & =\left[1–\left(3x–4y\right)\right]\left[1+\left(3x–4y\right)\right]\\ & =\left(1–3x+4y\right)\left(1+3x–4y\right)\end{array}$
109. $\begin{array}{cccc}1+11x+24{x}^{2}& =1+3x+8x+24{x}^{2}& & .\begin{array}{c}24\\ 12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ \end{array}\\ & =\left(1+3x\right)+8x\left(1+3x\right)& & \\ & =\left(1+3x\right)\left(1+8x\right)& & \end{array}$
110. $9{x}^{2}{y}^{3}–27{x}^{3}{y}^{3}–9{x}^{5}{y}^{3}=9{x}^{2}{y}^{3}\left(1–3x–{x}^{2}\right)$
111. $\begin{array}{cc}{\left({a}^{2}+{b}^{2}–{c}^{2}\right)}^{2}–9{x}^{2}{y}^{2}& =\left[\left({a}^{2}+{b}^{2}–{c}^{2}\right)–3xy\right]\left[\left({a}^{2}+{b}^{2}–{c}^{2}\right)+3xy\right]\\ & =\left({a}^{2}+{b}^{2}–{c}^{2}–3xy\right)\left({a}^{2}+{b}^{2}–{c}^{2}+3xy\right)\end{array}$
112. $\begin{array}{cc}8{\left(a+1\right)}^{3}–1& =\left[2\left(a+1\right)–1\right]\left\{{\left[2\left(a+1\right)\right]}^{2}+2\left(a+1\right)+1\right\}\\ & =\left[2a+2–1\right]\left\{4{\left(a+1\right)}^{2}+2a+2+1\right\}\\ & =\left(2a+1\right)\left\{4\left({a}^{2}+2a+1\right)+2a+3\right\}\\ & =\left(2a+1\right)\left\{4{a}^{2}+8a+4+2a+3\right\}\\ & =\left(2a+1\right)\left(4{a}^{2}+10a+7\right)\end{array}$
113. $100{x}^{4}{y}^{6}–121{m}^{4}=\left(10{x}^{2}{y}^{3}–11{m}^{2}\right)\left(10{x}^{2}{y}^{3}+11{m}^{2}\right)$
114. $\begin{array}{cccc}{\left({a}^{2}+1\right)}^{2}+5\left({a}^{2}+1\right)–24& ={\left({a}^{2}+1\right)}^{2}–3\left({a}^{2}+1\right)+8\left({a}^{2}+1\right)–24& & .\begin{array}{c}24\\ 12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ \end{array}\\ & =\left({a}^{2}+1\right)\left[\left({a}^{2}+1\right)–3\right]+8\left[\left({a}^{2}+1\right)–3\right]& & \\ & =\left[\left({a}^{2}+1\right)–3\right]\left[\left({a}^{2}+1\right)+8\right]& & \\ & =\left[{a}^{2}+1–3\right]\left[{a}^{2}+1+8\right]& & \\ & =\left({a}^{2}–2\right)\left({a}^{2}+9\right)& & \end{array}$
115. $\begin{array}{cc}1+1000{x}^{6}& =\left(1+10{x}^{2}\right)\left(1–10{x}^{2}+{\left[10{x}^{2}\right]}^{2}\right)\\ & =\left(1+10{x}^{2}\right)\left(1–10{x}^{2}+100{x}^{4}\right)\end{array}$
116. $\begin{array}{cc}49{a}^{2}–{x}^{2}–9{y}^{2}+6xy& =49{a}^{2}–{x}^{2}+6xy–9{y}^{2}\\ & =49{a}^{2}–\left({x}^{2}–6xy+9{y}^{2}\right)\\ & =49{a}^{2}–{\left(x–3y\right)}^{2}\\ & =\left[7a–\left(x–3y\right)\right]\left[7a+\left(x–3y\right)\right]\\ & =\left(7a–x+3y\right)\left(7a+x–3y\right)\end{array}$
117. $\begin{array}{cc}{x}^{4}–{y}^{2}+4{x}^{2}+4–4yz–4{z}^{2}& ={x}^{4}+4{x}^{2}+4–{y}^{2}–4yz–4{z}^{2}\\ & =\left({x}^{4}+4{x}^{2}+4\right)–\left({y}^{2}+4yz+4{z}^{2}\right)\\ & ={\left({x}^{2}+2\right)}^{2}–{\left(y+2z\right)}^{2}\\ & =\left[\left({x}^{2}+2\right)–\left(y+2z\right)\right]\left[\left({x}^{2}+2\right)+\left(y+2z\right)\right]\\ & =\left[{x}^{2}+2–y–2z\right]\left[{x}^{2}+2+y+2z\right]\\ & =\left({x}^{2}–y–2z+2\right)\left({x}^{2}+y+2z+2\right)\end{array}$
118. $\begin{array}{cc}{a}^{3}–64& =\left(a–4\right)\left({a}^{2}+4a+{4}^{2}\right)\\ & =\left(a–4\right)\left({a}^{2}+4a+16\right)\end{array}$
119. ${a}^{5}+{x}^{5}=\left(a+x\right)\left({a}^{4}–{a}^{3}x+{a}^{2}{x}^{2}–a{x}^{3}+{x}^{4}\right)$
120. $\begin{array}{cccc}{a}^{6}–3{a}^{3}b–54{b}^{2}& ={a}^{6}+6{a}^{3}b–9{a}^{3}b–54{b}^{2}& & .\begin{array}{c}54\\ 27\\ 9\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 9\\ \end{array}\\ & ={a}^{3}\left({a}^{3}+6b\right)–9b\left({a}^{3}+6b\right)& & \\ & =\left({a}^{3}+6b\right)\left({a}^{3}–9b\right)& & \end{array}$
121. $\begin{array}{cccc}165+4x–{x}^{2}& =165–11x+15x–{x}^{2}& & .\begin{array}{c}165\\ 55\\ 11\\ 1\end{array}|\begin{array}{c}3\\ 5\\ 11\\ \end{array}\\ & =11\left(15–x\right)+x\left(15–x\right)& & \\ & =\left(15–x\right)\left(11+x\right)& & \end{array}$
122. $\begin{array}{cc}{a}^{4}+{a}^{2}+1& ={a}^{4}+{a}^{2}+1+{a}^{2}–{a}^{2}\\ & ={a}^{4}+2{a}^{2}+1–{a}^{2}\\ & ={\left({a}^{2}+1\right)}^{2}–{a}^{2}\\ & =\left[\left({a}^{2}+1\right)–a\right]\left[\left({a}^{2}+1\right)+a\right]\\ & =\left({a}^{2}–a+1\right)\left({a}^{2}+a+1\right)\end{array}$
123. $\frac{{x}^{2}}{4}–\frac{{y}^{6}}{81}=\left(\frac{x}{2}–\frac{{y}^{3}}{9}\right)\left(\frac{x}{2}+\frac{{y}^{3}}{9}\right)$
124. $16{x}^{2}+\frac{8xy}{5}+\frac{{y}^{2}}{25}={\left(4x+\frac{y}{5}\right)}^{2}$
125. $\begin{array}{cccc}{a}^{4}{b}^{4}+4{a}^{2}{b}^{2}–96& ={a}^{4}{b}^{4}–8{a}^{2}{b}^{2}+12{a}^{2}{b}^{2}–96& & .\begin{array}{c}96\\ 48\\ 24\\ 12\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 12\\ \end{array}\\ & ={a}^{2}{b}^{2}\left({a}^{2}{b}^{2}–8\right)+12\left({a}^{2}{b}^{2}–8\right)& & \\ & =\left({a}^{2}{b}^{2}–8\right)\left({a}^{2}{b}^{2}+12\right)& & \end{array}$
126. $\begin{array}{cc}8{a}^{2}x+7y+21by–7ay–8{a}^{3}x+24{a}^{2}bx& =8{a}^{2}x–8{a}^{3}x+24{a}^{2}bx+7y–7ay+21by\\ & =8{a}^{2}x\left(1–a+3b\right)+7y\left(1–a+3b\right)\\ & =\left(1–a+3b\right)\left(8{a}^{2}x+7y\right)\end{array}$
127. $\begin{array}{cccc}{x}^{4}+11{x}^{2}–390& ={x}^{4}–15{x}^{2}+26{x}^{2}–390& & .\begin{array}{c}390\\ 195\\ 65\\ 13\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ 13\\ \end{array}\\ & ={x}^{2}\left({x}^{2}–15\right)+26\left({x}^{2}–15\right)& & \\ & =\left({x}^{2}–15\right)\left({x}^{2}+26\right)& & \end{array}$
128. $\begin{array}{cccc}7+33m–10{m}^{2}& =7–2m+35m–10{m}^{2}& & .\begin{array}{c}70\\ 35\\ 1\end{array}|\begin{array}{c}2\\ 35\\ \end{array}\\ & =\left(7–2m\right)+5m\left(7–2m\right)& & \\ & =\left(7–2m\right)\left(1+5m\right)& & \end{array}$
129. $\begin{array}{cc}4{\left(a+b\right)}^{2}–9{\left(c+d\right)}^{2}& =\left[2\left(a+b\right)–3\left(c+d\right)\right]\left[2\left(a+b\right)+3\left(c+d\right)\right]\\ & =\left(2a+2b–3c–3d\right)\left(2a+2b+3c+3d\right)\end{array}$
130. $\begin{array}{cc}729–125{x}^{3}{y}^{12}& =\left(9–5x{y}^{4}\right)\left[{9}^{2}+9\left(5x{y}^{4}\right)+{\left(5x{y}^{4}\right)}^{2}\right]\\ & =\left(9–5x{y}^{4}\right)\left(81+45x{y}^{4}+25{x}^{2}{y}^{8}\right)\end{array}$
131. $\begin{array}{cc}{\left(x+y\right)}^{2}+x+y& ={\left(x+y\right)}^{2}+\left(x+y\right)\\ & =\left(x+y\right)\left[\left(x+y\right)+1\right]\\ & =\left(x+y\right)\left(x+y+1\right)\end{array}$
132. $\begin{array}{cc}4–\left({a}^{2}+{b}^{2}\right)+2ab& =4–{a}^{2}–{b}^{2}+2ab\\ & =4–{a}^{2}+2ab–{b}^{2}\\ & =4–\left({a}^{2}–2ab+{b}^{2}\right)\\ & =4–{\left(a–b\right)}^{2}\\ & =\left[2–\left(a–b\right)\right]\left[2+\left(a–b\right)\right]\\ & =\left(2–a+b\right)\left(2+a–b\right)\end{array}$
133. $\begin{array}{cc}{x}^{3}–{y}^{3}+x–y& =\left({x}^{3}–{y}^{3}\right)+x–y\\ & =\left(x–y\right)\left({x}^{2}+xy+{y}^{2}\right)+\left(x–y\right)\\ & =\left(x–y\right)\left[\left({x}^{2}+xy+{y}^{2}\right)+1\right]\\ & =\left(x–y\right)\left({x}^{2}+xy+{y}^{2}+1\right)\end{array}$
134. $\begin{array}{cc}{a}^{2}–{b}^{2}+{a}^{3}–{b}^{3}& =\left({a}^{2}–{b}^{2}\right)+\left({a}^{3}–{b}^{3}\right)\\ & =\left(a–b\right)\left(a+b\right)+\left(a–b\right)\left({a}^{2}+ab+{b}^{2}\right)\\ & =\left(a–b\right)\left[\left(a+b\right)+\left({a}^{2}+ab+{b}^{2}\right)\right]\\ & =\left(a–b\right)\left(a+b+{a}^{2}+ab+{b}^{2}\right)\end{array}$

## Ejercicio 105

CAPITULO X

Descomposición Factorial
Ejercicio 105
Factorar:
1. ${a}^{5}+1=\left(a+1\right)\left({a}^{4}–{a}^{3}+{a}^{2}–a+1\right)$
2. ${a}^{5}–1=\left(a–1\right)\left({a}^{4}+{a}^{3}+{a}^{2}+a+1\right)$
3. $1–{x}^{5}=\left(1–x\right)\left(1+x+{x}^{2}+{x}^{3}+{x}^{4}\right)$
4. ${a}^{7}+{b}^{7}=\left(a+b\right)\left({a}^{6}–{a}^{5}b+{a}^{4}{b}^{2}–{a}^{3}{b}^{3}+{a}^{2}{b}^{4}–a{b}^{5}+{b}^{6}\right)$
5. ${m}^{7}–{n}^{7}=\left(m–n\right)\left({m}^{6}+{m}^{5}n+{m}^{4}{n}^{2}+{m}^{3}{n}^{3}+{m}^{2}{n}^{4}+m{n}^{5}+{n}^{6}\right)$
6. ${a}^{5}+243=\left(a+3\right)\left({a}^{4}–3{a}^{3}+9{a}^{2}–27a+81\right)$
7. $32–{m}^{5}=\left(2–m\right)\left(16+8m+4{m}^{2}+2{m}^{3}+{m}^{4}\right)$
8. $1+243{x}^{5}=\left(1+3x\right)\left(1–3x+9{x}^{2}–27{x}^{3}+81{x}^{4}\right)$
9. ${x}^{7}+128=\left(x+2\right)\left({x}^{6}–2{x}^{5}+4{x}^{4}–8{x}^{3}+16{x}^{2}–32x+64\right)$
10. $243–32{b}^{5}=\left(3–2b\right)\left(81+54b+36{b}^{2}+24{b}^{3}+16{b}^{4}\right)$
11. ${a}^{5}+{b}^{5}{c}^{5}=\left(a+bc\right)\left({a}^{4}–{a}^{3}bc+{a}^{2}{b}^{2}{c}^{2}–a{b}^{3}{c}^{3}+{b}^{4}{c}^{4}\right)$
12. ${m}^{7}–{a}^{7}{x}^{7}=\left(m–ax\right)\left({m}^{6}+{m}^{5}ax+{m}^{4}{a}^{2}{x}^{2}+{m}^{3}{a}^{3}{x}^{3}+{m}^{2}{a}^{4}{x}^{4}+m{a}^{5}{x}^{5}+{a}^{6}{x}^{6}\right)$
13. $1+{x}^{7}=\left(1+x\right)\left(1–x+{x}^{2}–{x}^{3}+{x}^{4}–{x}^{5}+{x}^{6}\right)$
14. ${x}^{7}–{y}^{7}=\left(x–y\right)\left({x}^{6}+{x}^{5}y+{x}^{4}{y}^{2}+{x}^{3}{y}^{3}+{x}^{2}{y}^{4}+x{y}^{5}+{y}^{6}\right)$
15. ${a}^{7}+2187=\left(a+3\right)\left({a}^{6}–3{a}^{5}+9{a}^{4}–27{a}^{3}+81{a}^{2}–243a+729\right)$
16. $1–128{a}^{7}=\left(1–2a\right)\left(1+2a+4{a}^{2}+8{a}^{3}+16{a}^{4}+32{a}^{5}+64{a}^{6}\right)$
17. ${x}^{10}+32{y}^{5}=\left({x}^{2}+2y\right)\left({x}^{8}–2{x}^{6}y+4{x}^{4}{y}^{2}–8{x}^{2}{y}^{3}+16{y}^{4}\right)$
18. $1+128{x}^{14}=\left(1+2{x}^{2}\right)\left(1–2{x}^{2}+4{x}^{4}–8{x}^{6}+16{x}^{8}–32{x}^{10}+64{x}^{12}\right)$

## Ejercicio 104

CAPITULO X

Descomposición Factorial
Ejercicio 104
Descomponer en 2 factores:
1. $\begin{array}{cc}1+{\left(x+y\right)}^{3}& =\left[1+\left(x+y\right)\right]\left[1–\left(x+y\right)+{\left(x+y\right)}^{2}\right]\\ & =\left(x+y+1\right)\left(1–x–y+{x}^{2}+2xy+{y}^{2}\right)\end{array}$
2. $\begin{array}{cc}1–{\left(a+b\right)}^{3}& =\left[1–\left(a+b\right)\right]\left[1+\left(a+b\right)+{\left(a+b\right)}^{2}\right]\\ & =\left(1–a–b\right)\left(1+a+b+{a}^{2}+2ab+{b}^{2}\right)\end{array}$
3. $\begin{array}{cc}27+{\left(m–n\right)}^{3}& =\left[3+\left(m–n\right)\right]\left[{3}^{2}–3\left(m–n\right)+{\left(m–n\right)}^{2}\right]\\ & =\left(3+m–n\right)\left(9–3m+3n+{m}^{2}–2mn+{n}^{2}\right)\end{array}$
4. $\begin{array}{cc}{\left(x–y\right)}^{3}–8& =\left[\left(x–y\right)–2\right]\left[{\left(x–y\right)}^{2}+2\left(x–y\right)+4\right]\\ & =\left(x–y–2\right)\left({x}^{2}–2xy+{y}^{2}+2x–2y+4\right)\end{array}$
5. $\begin{array}{cc}{\left(x+2y\right)}^{3}+1& =\left[\left(x+2y\right)+1\right]\left[{\left(x+2y\right)}^{2}–\left(x+2y\right)+1\right]\\ & =\left(x+2y+1\right)\left({x}^{2}+4xy+4{y}^{2}–x–2y+1\right)\end{array}$
6. $\begin{array}{cc}1–{\left(2a–b\right)}^{3}& =\left[1–\left(2a–b\right)\right]\left[1+\left(2a–b\right)+{\left(2a–b\right)}^{2}\right]\\ & =\left(1–2a+b\right)\left(1+2a–b+4{a}^{2}–4ab+{b}^{2}\right)\end{array}$
7. $\begin{array}{cc}{a}^{3}+{\left(a+1\right)}^{3}& =\left[a+\left(a+1\right)\right]\left[{a}^{2}–a\left(a+1\right)+{\left(a+1\right)}^{2}\right]\\ & =\left(2a+1\right)\left(\overline{){a}^{2}}–\overline{){a}^{2}}–a+{a}^{2}+2a+1\right)\\ & =\left(2a+1\right)\left({a}^{2}–a+1\right)\end{array}$
8. $\begin{array}{cc}8{a}^{3}–{\left(a–1\right)}^{3}& =\left[2a–\left(a–1\right)\right]\left[{\left(2a\right)}^{2}+2a\left(a–1\right)+{\left(a–1\right)}^{2}\right]\\ & =\left(2a–a+1\right)\left(4{a}^{2}+2{a}^{2}–2a+{a}^{2}–2a+1\right)\\ & =\left(a+1\right)\left(7{a}^{2}–4a+1\right)\end{array}$
9. $\begin{array}{cc}27{x}^{3}–{\left(x–y\right)}^{3}& =\left[3x–\left(x–y\right)\right]\left[{\left(3x\right)}^{2}+3x\left(x–y\right)+{\left(x–y\right)}^{2}\right]\\ & =\left(3x–x+y\right)\left(9{x}^{2}+3{x}^{2}–3xy+{x}^{2}–2xy+{y}^{2}\right)\\ & =\left(2x+y\right)\left(13{x}^{2}–5xy+{y}^{2}\right)\end{array}$
10. $\begin{array}{cc}{\left(2a–b\right)}^{3}–27& =\left[\left(2a–b\right)–3\right]\left[{\left(2a–b\right)}^{2}+3\left(2a–b\right)+{3}^{2}\right]\\ & =\left(2a–b–3\right)\left(4{a}^{2}–4ab+{b}^{2}+6a–3b+9\right)\end{array}$
11. $\begin{array}{cc}{x}^{6}–{\left(x+2\right)}^{3}& =\left[{x}^{2}–\left(x+2\right)\right]\left[{\left({x}^{2}\right)}^{2}+{x}^{2}\left(x+2\right)+{\left(x+2\right)}^{2}\right]\\ & =\left({x}^{2}–x–2\right)\left({x}^{4}+{x}^{4}+2{x}^{2}+{x}^{2}+4x+4\right)\\ & =\left({x}^{2}–x–2\right)\left(2{x}^{4}+3{x}^{2}+4x+4\right)\end{array}$
12. $\begin{array}{cc}{\left(a+1\right)}^{3}+{\left(a–3\right)}^{3}& =\left[\left(a+1\right)+\left(a–3\right)\right]\left[{\left(a+1\right)}^{2}–\left(a+1\right)\left(a–3\right)+{\left(a–3\right)}^{2}\right]\\ & =\left(a+1+a–3\right)\left[{a}^{2}+2a+1–\left({a}^{2}–2a–3\right)+{a}^{2}–6a+9\right]\\ & =\left(2a–2\right)\left[2{a}^{2}–4a+10–{a}^{2}+2a+3\right]\\ & =2\left(a–1\right)\left({a}^{2}–2a+13\right)\end{array}$
13. $\begin{array}{cc}{\left(x–1\right)}^{3}–{\left(x+2\right)}^{3}& =\left[\left(x–1\right)–\left(x+2\right)\right]\left[{\left(x–1\right)}^{2}+\left(x–1\right)\left(x+2\right)+{\left(x+2\right)}^{2}\right]\\ & =\left[\overline{)x}–1–\overline{)x}–2\right]\left[{x}^{2}–2x+1+{x}^{2}+x–2+{x}^{2}+4x+4\right]\\ & =–3\left(3{x}^{2}+3x+3\right)\\ & =–9\left({x}^{2}+x+1\right)\end{array}$
14. $\begin{array}{cc}{\left(x–y\right)}^{3}–{\left(x+y\right)}^{3}& =\left[\left(x–y\right)–\left(x+y\right)\right]\left[{\left(x–y\right)}^{2}+\left(x–y\right)\left(x+y\right)+{\left(x+y\right)}^{2}\right]\\ & =\left[\overline{)x}–y–\overline{)x}–y\right]\left[{x}^{2}–\overline{)2xy}+\overline{){y}^{2}}+{x}^{2}–\overline{){y}^{2}}+{x}^{2}+\overline{)2xy}+{y}^{2}\right]\\ & =–2y\left(3{x}^{2}+{y}^{2}\right)\end{array}$
15. $\begin{array}{cc}{\left(m–2\right)}^{3}+{\left(m–3\right)}^{3}& =\left[\left(m–2\right)+\left(m–3\right)\right]\left[{\left(m–2\right)}^{2}–\left(m–2\right)\left(m–3\right)+{\left(m–3\right)}^{2}\right]\\ & =\left[m–2+m–3\right]\left[{m}^{2}–4m+4–\left({m}^{2}–5m+6\right)+{m}^{2}–6m+9\right]\\ & =\left(2m–5\right)\left[2{m}^{2}–10m+13–{m}^{2}+5m–6\right]\\ & =\left(2m–5\right)\left({m}^{2}–5m+7\right)\end{array}$
16. $\begin{array}{cc}{\left(2x–y\right)}^{3}+{\left(3x+y\right)}^{3}& =\left[\left(2x–y\right)+\left(3x+y\right)\right]\left[{\left(2x–y\right)}^{2}–\left(2x–y\right)\left(3x+y\right)+{\left(3x+y\right)}^{2}\right]\\ & =\left[2x–\overline{)y}+3x+\overline{)y}\right]\left[4{x}^{2}–4xy+{y}^{2}–\left(6{x}^{2}–xy–{y}^{2}\right)+9{x}^{2}+6xy+{y}^{2}\right]\\ & =5x\left[13{x}^{2}+2xy+2{y}^{2}–6{x}^{2}+xy+{y}^{2}\right]\\ & =5x\left(7{x}^{2}+3xy+3{y}^{2}\right)\end{array}$
17. $\begin{array}{cc}8{\left(a+b\right)}^{3}+{\left(a–b\right)}^{3}& =\left[2\left(a+b\right)+\left(a–b\right)\right]\left\{{\left[2\left(a+b\right)\right]}^{2}–2\left(a+b\right)\left(a–b\right)+{\left(a–b\right)}^{2}\right\}\\ & =\left[2a+2b+a–b\right]\left\{4{\left(a+b\right)}^{2}–2\left({a}^{2}–{b}^{2}\right)+{a}^{2}–2ab+{b}^{2}\right\}\\ & =\left(3a+b\right)\left\{4\left({a}^{2}+2ab+{b}^{2}\right)–2{a}^{2}+2{b}^{2}+{a}^{2}–2ab+{b}^{2}\right\}\\ & =\left(3a+b\right)\left\{4{a}^{2}+8ab+4{b}^{2}–{a}^{2}+3{b}^{2}–2ab\right\}\\ & =\left(3a+b\right)\left(3{a}^{2}+6ab+7{b}^{2}\right)\end{array}$
18. $\begin{array}{cc}64{\left(m+n\right)}^{3}–125& =\left[4\left(m+n\right)–5\right]\left\{{\left[4\left(m+n\right)\right]}^{2}+20\left(m+n\right)+{5}^{2}\right\}\\ & =\left[4m+4n–5\right]\left\{16{\left(m+n\right)}^{2}+20m+20n+25\right\}\\ & =\left(4m+4n–5\right)\left\{16\left({m}^{2}+2mn+{n}^{2}\right)+20m+20n+25\right\}\\ & =\left(4m+4n–5\right)\left(16{m}^{2}+32mn+16{n}^{2}+20m+20n+25\right)\end{array}$

## Ejercicio 103

CAPITULO X

Descomposición Factorial
Ejercicio 103
Descomponer en 2 factores:
1. $1+{a}^{3}=\left(1+a\right)\left(1–a+{a}^{2}\right)$
2. $1–{a}^{3}=\left(1–a\right)\left(1+a+{a}^{2}\right)$
3. ${x}^{3}+{y}^{3}=\left(x+y\right)\left({x}^{2}–xy+{y}^{2}\right)$
4. ${m}^{3}–{n}^{3}=\left(m–n\right)\left({m}^{2}+mn+{n}^{2}\right)$
5. ${a}^{3}–1=\left(a–1\right)\left({a}^{2}+a+1\right)$
6. ${y}^{3}+1=\left(y+1\right)\left({y}^{2}–y+1\right)$
7. ${y}^{3}–1=\left(y–1\right)\left({y}^{2}+y+1\right)$
8. $8{x}^{3}–1=\left(2x–1\right)\left(4{x}^{2}+2x+1\right)$
9. $1–8{x}^{3}=\left(1–2x\right)\left(1+2x+4{x}^{2}\right)$
10. ${x}^{3}–27=\left(x–3\right)\left({x}^{2}+3x+9\right)$
11. ${a}^{3}+27=\left(a+3\right)\left({a}^{2}–3a+9\right)$
12. $8{x}^{3}+{y}^{3}=\left(2x+y\right)\left(4{x}^{2}–2xy+{y}^{2}\right)$
13. $27{a}^{3}–{b}^{3}=\left(3a–b\right)\left(9{a}^{2}+3ab+{b}^{2}\right)$
14. $64+{a}^{6}=\left(4+{a}^{2}\right)\left(16–4{a}^{2}+{a}^{4}\right)$
15. ${a}^{3}–125=\left(a–5\right)\left({a}^{2}+5a+25\right)$
16. $1–216{m}^{3}=\left(1–6m\right)\left(1+6m+36{m}^{2}\right)$
17. $8{a}^{3}+27{b}^{6}=\left(2a+3{b}^{2}\right)\left(4{a}^{2}–6a{b}^{2}+9{b}^{6}\right)$
18. ${x}^{6}–{b}^{9}=\left({x}^{2}–{b}^{3}\right)\left({x}^{4}+{x}^{2}{b}^{3}+{b}^{3}\right)$
19. $8{x}^{3}–27{y}^{3}=\left(2x–3y\right)\left(4{x}^{2}+6xy+9{y}^{2}\right)$
20. $1+343{n}^{3}=\left(1+7n\right)\left(1–7n+49{n}^{2}\right)$
21. $64{a}^{3}–729=\left(4a–9\right)\left(16{a}^{2}+36a+81\right)$
22. ${a}^{3}{b}^{3}–{x}^{6}=\left(ab–{x}^{2}\right)\left({a}^{2}{b}^{2}+ab{x}^{2}+{x}^{4}\right)$
23. $512+27{a}^{9}=\left(8+3{a}^{3}\right)\left(64–24{a}^{3}+9{a}^{6}\right)$
24. ${x}^{6}–8{y}^{12}=\left({x}^{2}–2{y}^{4}\right)\left({x}^{4}+2{x}^{2}{y}^{4}+4{y}^{8}\right)$
25. $1+729{x}^{6}=\left(1+9{x}^{2}\right)\left(1–9{x}^{2}+81{x}^{4}\right)$
26. $27{m}^{3}+64{n}^{9}=\left(3m+4{n}^{3}\right)\left(3{m}^{2}–12m{n}^{3}+16{n}^{6}\right)$
27. $343{x}^{3}+512{y}^{6}=\left(7x+8{y}^{2}\right)\left(49{x}^{2}–56x{y}^{2}+64{y}^{4}\right)$
28. ${x}^{3}{y}^{6}–216{y}^{9}=\left(x{y}^{2}–6{y}^{3}\right)\left({x}^{2}{y}^{4}+6x{y}^{5}+36{y}^{6}\right)$
29. ${a}^{3}{b}^{3}{x}^{3}+1=\left(abx+1\right)\left({a}^{2}{b}^{2}{x}^{2}–abx+1\right)$
30. ${x}^{9}+{y}^{9}=\left({x}^{3}+{y}^{3}\right)\left({x}^{6}–{x}^{3}{y}^{3}+{y}^{6}\right)$
31. $1000{x}^{3}–1=\left(10x–1\right)\left(100{x}^{2}+10x+1\right)$
32. ${a}^{6}+125{b}^{12}=\left({a}^{2}+5{b}^{4}\right)\left({a}^{4}–5{a}^{2}{b}^{4}+25{b}^{8}\right)$
33. ${x}^{12}+{y}^{12}=\left({x}^{4}+{y}^{4}\right)\left({x}^{8}–{x}^{4}{y}^{4}+{y}^{8}\right)$
34. $1–27{a}^{3}{b}^{3}=\left(1–3ab\right)\left(1+3ab+9{a}^{2}{b}^{2}\right)$
35. $8{x}^{6}+729=\left(2{x}^{2}+9\right)\left(4{x}^{4}–18{x}^{2}+81\right)$
36. ${a}^{3}+8{b}^{12}=\left(a+2{b}^{4}\right)\left({a}^{2}–2a{b}^{4}+4{b}^{8}\right)$
37. $8{x}^{9}–125{y}^{3}{z}^{6}=\left(2{x}^{3}–5y{z}^{2}\right)\left(4{x}^{6}+10{x}^{3}y{z}^{2}+25{y}^{2}{z}^{4}\right)$
38. $27{m}^{6}+343{n}^{9}=\left(3{m}^{2}+7{n}^{3}\right)\left(9{m}^{4}–21{m}^{2}{n}^{3}+49{n}^{6}\right)$
39. $216–{x}^{12}=\left(6–{x}^{4}\right)\left(36+6{x}^{4}+{x}^{8}\right)$

## Ejercicio 102

CAPITULO X

Descomposición Factorial
Ejercicio 102
Factorar por el método anterior, si es posible, las expresiones siguientes, ordenándolas previamente:
Sabemos que en los productos notables que:$\begin{array}{cc}{\left(a+b\right)}^{3}& =\left({a}^{3}+3{a}^{2}b+3a{b}^{2}+{b}^{3}\right)\\ {\left(a–b\right)}^{3}& =\left({a}^{3}–3{a}^{2}b+3a{b}^{2}–{b}^{3}\right)\end{array}$

## Ejercicio 101

CAPITULO X

Descomposición Factorial
Ejercicio 101
Factorar:
1. $\begin{array}{cccc}6{x}^{4}+5{x}^{2}–6& =6{x}^{4}+9{x}^{2}–4{x}^{2}–6& & .\begin{array}{c}36\\ 18\\ 9\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 9\\ \end{array}\\ & =3{x}^{2}\left(2{x}^{2}+3\right)–2\left(2{x}^{2}+3\right)& & \\ & =\left(2{x}^{2}+3\right)\left(3{x}^{2}–2\right)& & \end{array}$
2. $\begin{array}{cccc}5{x}^{6}+4{x}^{3}–12& =5{x}^{6}+10{x}^{3}–6{x}^{3}–12& & .\begin{array}{c}60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 5\\ \end{array}\\ & =5{x}^{3}\left({x}^{3}+2\right)–6\left({x}^{3}+2\right)& & \\ & =\left({x}^{3}+2\right)\left(5{x}^{3}–6\right)& & \end{array}$
3. $\begin{array}{cccc}10{x}^{8}+29{x}^{4}+10& =10{x}^{8}+25{x}^{4}+4{x}^{4}+10& & .\begin{array}{c}100\\ 50\\ 25\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 25\\ \end{array}\\ & =5{x}^{4}\left(2{x}^{2}+5\right)+2\left(2{x}^{2}+5\right)& & \\ & =\left(2{x}^{2}+5\right)\left(5{x}^{5}+2\right)& & \end{array}$
4. $\begin{array}{cccc}6{a}^{2}{x}^{2}+5ax–21& =6{a}^{2}{x}^{2}–9ax+14ax–21& & .\begin{array}{c}126\\ 63\\ 21\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 7\\ \end{array}\\ & =3ax\left(2ax–3\right)+7\left(2ax–3\right)& & \\ & =\left(2ax–3\right)\left(3ax+7\right)& & \end{array}$
5. $\begin{array}{cccc}20{x}^{2}{y}^{2}+9xy–20& =20{x}^{2}{y}^{2}+25xy–16xy–20& & .\begin{array}{c}400\\ 200\\ 100\\ 50\\ 25\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 2\\ 25\\ \end{array}\\ & =5xy\left(4xy+5\right)–4\left(4xy+5\right)& & \\ & =\left(4xy+5\right)\left(5xy–4\right)& & \end{array}$
6. $\begin{array}{cccc}15{x}^{2}–ax–2{a}^{2}& =15{x}^{2}+5ax–6ax–2{a}^{2}& & .\begin{array}{c}30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ \end{array}\\ & =5x\left(3x+a\right)–2a\left(3x+a\right)& & \\ & =\left(3x+a\right)\left(5x–2a\right)& & \end{array}$
7. $\begin{array}{cccc}12–7x–10{x}^{2}& =12+8x–15x–10{x}^{2}& & .\begin{array}{c}120\\ 60\\ 30\\ 15\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 15\\ \end{array}\\ & =4\left(3+2x\right)–5x\left(3+2x\right)& & \\ & =\left(3+2x\right)\left(4–5x\right)& & \end{array}$
8. $\begin{array}{cccc}21{x}^{2}–29xy–72{y}^{2}& =21{x}^{2}–56xy+27xy–72{y}^{2}& & .\begin{array}{c}1512\\ 756\\ 378\\ 189\\ 63\\ 21\\ 7\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ 3\\ 3\\ 7\\ \end{array}\\ & =7x\left(3x–8y\right)–9y\left(3x–8y\right)& & \\ & =\left(3x–8y\right)\left(7x–9y\right)& & \end{array}$
9. $\begin{array}{cccc}6{m}^{2}–13am–15{a}^{2}& =6{m}^{2}–18am+5am–15{a}^{2}& & .\begin{array}{c}90\\ 45\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 5\\ \end{array}\\ & =6m\left(m–3a\right)–5a\left(m–3a\right)& & \\ & =\left(m–3a\right)\left(6m–5a\right)& & \end{array}$
10. $\begin{array}{cccc}14{x}^{4}–45{x}^{2}–14& =14{x}^{4}–49{x}^{2}+4{x}^{2}–14& & .\begin{array}{c}196\\ 98\\ 49\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 49\\ \end{array}\\ & =7{x}^{2}\left(2{x}^{2}–7\right)–2\left(2{x}^{2}–7\right)& & \\ & =\left(2{x}^{2}–7\right)\left(7{x}^{2}–2\right)& & \end{array}$
11. $\begin{array}{cccc}30{a}^{2}–13ab–3{b}^{2}& =30{a}^{2}+5ab–18ab–3{b}^{2}& & .\begin{array}{c}90\\ 45\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 5\\ \end{array}\\ & =5a\left(6a+b\right)–3b\left(6a+b\right)& & \\ & =\left(6a+b\right)\left(5a–3b\right)& & \end{array}$
12. $\begin{array}{cccc}7{x}^{6}–33{x}^{3}–10& =7{x}^{6}–35{x}^{3}+2{x}^{3}–10& & .\begin{array}{c}70\\ 35\\ 1\end{array}|\begin{array}{c}2\\ 35\\ \end{array}\\ & =7{x}^{3}\left({x}^{3}–5\right)–2\left({x}^{3}–5\right)& & \\ & =\left({x}^{3}–5\right)\left(7{x}^{3}–2\right)& & \end{array}$
13. $\begin{array}{cccc}30+13a–3{a}^{2}& =30–5a+18a–3{a}^{2}& & .\begin{array}{c}90\\ 45\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 5\\ \end{array}\\ & =5\left(6–a\right)–3a\left(6–a\right)& & \\ & =\left(6–a\right)\left(5–3a\right)& & \end{array}$
14. $\begin{array}{cccc}5+7{x}^{4}–6{x}^{8}& =5+10{x}^{4}–3{x}^{4}–6{x}^{8}& & .\begin{array}{c}30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ \end{array}\\ & =5\left(1+2{x}^{4}\right)–3{x}^{4}\left(1+2{x}^{4}\right)& & \\ & =\left(1+2{x}^{4}\right)\left(5–3{x}^{4}\right)& & \end{array}$
15. $\begin{array}{cccc}6{a}^{2}–ax–15{x}^{2}& =6{a}^{2}+9ax–10ax–15{x}^{2}& & .\begin{array}{c}90\\ 45\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 5\\ \end{array}\\ & =3a\left(2a+3x\right)–5x\left(2a+3x\right)& & \\ & =\left(2a+3x\right)\left(3a–5\right)& & \end{array}$
16. $\begin{array}{cccc}4{x}^{2}+7mnx–15{m}^{2}{n}^{2}& =4{x}^{2}+12mnx–5mnx–15{m}^{2}{n}^{2}& & .\begin{array}{c}60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 5\\ \end{array}\\ & =4x\left(x+3mn\right)–5mn\left(x+3mn\right)& & \\ & =\left(x+3mn\right)\left(4x–5mn\right)& & \end{array}$
17. $\begin{array}{cccc}18{a}^{2}+17ay–15{y}^{2}& =18{a}^{2}+27ay–10ay–15{y}^{2}& & .\begin{array}{c}270\\ 135\\ 45\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 3\\ 5\\ \end{array}\\ & =9a\left(2a+3y\right)–5y\left(2a+3y\right)& & \\ & =\left(2a+3y\right)\left(9a–5\right)& & \end{array}$
18. $\begin{array}{cccc}15+2{x}^{2}–8{x}^{4}& =15–10{x}^{2}+12{x}^{2}–8{x}^{4}& & .\begin{array}{c}120\\ 60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ 5\\ \end{array}\\ & =5\left(3–2{x}^{2}\right)+4{x}^{2}\left(3–2{x}^{2}\right)& & \\ & =\left(3–2{x}^{2}\right)\left(5+4{x}^{2}\right)& & \end{array}$
19. $\begin{array}{cccc}6–25{x}^{8}+5{x}^{4}=6+5{x}^{4}–25{x}^{8}& =6–10{x}^{4}+15{x}^{4}–25{x}^{8}& & .\begin{array}{c}150\\ 75\\ 25\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 5\\ 5\\ \end{array}\\ & =2\left(3–5{x}^{4}\right)+5{x}^{4}\left(3–5{x}^{4}\right)& & \\ & =\left(3–5{x}^{4}\right)\left(2+5{x}^{4}\right)& & \end{array}$
20. $\begin{array}{cccc}30{x}^{10}–91{x}^{5}–30& =30{x}^{10}+100{x}^{5}–9{x}^{5}–30& & .\begin{array}{c}900\\ 450\\ 225\\ 75\\ 25\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 3\\ 5\\ 5\\ \end{array}\\ & =10{x}^{5}\left(3{x}^{5}–10\right)–3\left(3{x}^{5}–10\right)& & \\ & =\left(3{x}^{5}–10\right)\left(10{x}^{5}–3\right)& & \end{array}$
21. $\begin{array}{cccc}30{m}^{2}+17am–21{a}^{2}& =30{m}^{2}+35am–18am–21{a}^{2}& & .\begin{array}{c}630\\ 315\\ 105\\ 35\\ 1\end{array}|\begin{array}{c}2\\ 3\\ 3\\ 35\\ \end{array}\\ & =5m\left(6m+7a\right)–3a\left(6m+7a\right)& & \\ & =\left(6m+7a\right)\left(5m–3a\right)& & \end{array}$
22. $\begin{array}{cccc}16a–4–15{a}^{2}& =–\left(15{a}^{2}–16a+4\right)& & .\begin{array}{c}60\\ 30\\ 15\\ 5\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 5\\ \end{array}\\ & =–\left(15{a}^{2}–10a–6a+4\right)& & \\ & =–\left[5a\left(3a–2\right)–2\left(3a–2\right)\right]& & \\ & =–\left(3a–2\right)\left(5a–2\right)& & \\ & =\left(2–3a\right)\left(5a–2\right)& & \end{array}$
23. $\begin{array}{cccc}11xy–6{y}^{2}–4{x}^{2}& =–\left(4{x}^{2}–11xy+6{y}^{2}\right)& & .\begin{array}{c}24\\ 12\\ 6\\ 3\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 2\\ 3\\ \end{array}\\ & =–\left(4{x}^{2}–8xy–3xy+6{y}^{2}\right)& & \\ & =–\left[4x\left(x–2y\right)–3y\left(x–2y\right)\right]& & \\ & =–\left(x–2y\right)\left(4x–3y\right)& & \\ & =\left(2y–x\right)\left(4x–3y\right)& & \end{array}$
24. $\begin{array}{cccc}27ab–9{b}^{2}–20{a}^{2}& =–\left(20{a}^{2}–27ab+9{b}^{2}\right)& & .\begin{array}{c}180\\ 90\\ 45\\ 15\\ 1\end{array}|\begin{array}{c}2\\ 2\\ 3\\ 15\\ \end{array}\\ & =–\left(20{a}^{2}–15ab–12ab+9{b}^{2}\right)& & \\ & =–\left[5a\left(4a–3b\right)–3b\left(4a–3b\right)\right]& & \\ & =–\left(4a–3b\right)\left(5a–3b\right)& & \\ & =\left(3b–4a\right)\left(5a–3b\right)& & \end{array}$