Ejercicio 137

Ejercicio 137

1. a– a b b– 1 b = ab–a b b 2 –1 b = a( b–1 ) b 2 –1 = a ( b–1 ) ( b–1 ) ( b+1 ) = a b+1
2. x 2 – 1 x 1– 1 x = x 3 –1 x x–1 x = ( x–1 ) ( x 2 +x+1 ) x–1 = x 2 +x+1
3. a b – b a 1+ b a = a 2 – b 2 a b a+b a = a 2 – b 2 b( a+b ) = ( a–b ) ( a+b ) b ( a+b ) = a–b b
4. 1 m + 1 n 1 m – 1 n = n+m mn n–m mn = n+m n–m
5. x+ x 2 x– x 4 = 2x+x 2 4x–x = 2 ( 3x ) 3x =2
6. x y – y x 1+ y x = x 2 – y 2 x y x+y x = x 2 – y 2 y( x+y ) = ( x–y ) ( x+y ) y ( x+y ) = x–y y
7. x+4+ 3 x x–4– 5 x = x( x+4 ) +3 x x( x–4 ) –5 x = x 2 +4x+3 x 2 –4x–5 = ( x+1 ) ( x+3 ) ( x+5 ) ( x+1 ) = x+3 x+5
8. a–4+ 4 a 1– 2 a = a( a–4 ) +4 a a–2 a = a 2 –4a+4 a–2 = ( a–2 ) 2 a–2 =a–2
9. 2 a 2 – b 2 a –b 4 a 2 + b 2 4ab +1 = 2 a 2 – b 2 –ab a 4 a 2 + b 2 +4ab 4 a b = 2 a 2 –ab– b 2 4 a 2 +4ab+ b 2 = 2 a 2 –2ab+ab– b 2 ( 2a+b ) 2 = 2a( a–b ) +b( a–b ) ( 2a+b ) 2 = ( 2a+b ) ( a–b ) ( 2a+b ) 2 = a–b 2a+b
10. 2+ 3a 5b a+ 10b 3 = 10b+3a 5b 3a+10b 3 = 3 5b
11. a–x+ x 2 a+x a 2 – a 2 a+x = ( a–x ) ( a+x ) + x 2 a+x a 2 ( a+x ) – a 2 a+x = a 2 – x 2 + x 2 a 2 [ ( a+x ) –1 ] = a 2 a 2 ( a+x–1 ) = 1 a+x–1
12. a+5– 14 a 1+ 8 a + 7 a 2 = a( a+5 ) –14 a a 2 +8a+7 a 2 = a( a 2 +5a–14 ) ( a+7 ) ( a+1 ) = a ( a+7 ) ( a–2 ) ( a+7 ) ( a+1 ) = a( a–2 ) a+1
13. 1 a – 9 a 2 + 20 a 3 16 a –a = a 2 –9a+20 a 16– a 2 a = a 2 –9a+20 a 2 ( 16– a 2 ) = ( a–5 ) ( a–4 ) a 2 ( 4–a ) ( 4+a ) =– ( a–5 ) ( a–4 ) a 2 ( a–4 ) ( 4+a ) = 5–a a 2 ( a+4 )
14. 20 x 2 +7x–6 x 4 x 2 –25 = 20 x 2 +15x–8x–6 x 4 x 2 –25 x 2 = x[ 5x( 4x+3 ) –2( 4x+3 ) ] 4 x 2 –25 = x( 5x–2 ) ( 4x+3 ) ( 2x–5 ) ( 2x+5 ) =– x ( 2x+5 ) ( 4x+3 ) ( 2x–5 ) ( 2x+5 ) = x( 4x+3 ) 5–2x
15. 1+ 1 x–1 1+ 1 x 2 –1 = x– 1 + 1 x–1 x 2 – 1 + 1 x 2 –1 = x ( x 2 –1 ) x 2 ( x–1 ) = ( x+1 ) ( x–1 ) x ( x–1 ) = x+1 x
16. a– ab a+b a+ ab a–b = a( a+b ) –ab a+b a( a–b ) +ab a–b = a[ ( a+b ) –b ] a+b a[ ( a–b ) +b ] a–b = a ( a+ b – b ) a+b a ( a– b + b ) a–b = a ( a–b ) a ( a+b ) = a–b a+b
17. x–1– 5 x+3 x+5– 35 x+3 = ( x–1 ) ( x+3 ) –5 x+3 ( x+5 ) ( x+3 ) –35 x+3 = x 2 +2x–3–5 x 2 +8x+15–35 = x 2 +2x–8 x 2 +8x–20 = ( x+4 ) ( x–2 ) ( x+10 ) ( x–2 ) = x+4 x+10
18. a+2– 7a+9 a+3 a–4+ 5a–11 a+1 = ( a+2 ) ( a+3 ) –( 7a+9 ) a+3 ( a–4 ) ( a+1 ) +5a–11 a+1 = a 2 +5a+6–7a–9 a+3 a 2 –3a–4+5a–11 a+1 = a 2 –2a–3 a+3 a 2 +2a–15 a+1 = ( a+1 ) ( a 2 –2a–3 ) ( a+3 ) ( a 2 +2a–15 ) = ( a+1 ) ( a–3 ) ( a+1 ) ( a+3 ) ( a+5 ) ( a–3 ) = ( a+1 ) 2 ( a+3 ) ( a+5 )