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CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES

Ejercicio 68

Miscelanea

Escribir, por simple inspección, el resultado de:

$\begin{matrix} {(x + 2)}^{2} & = {(x)}^{2} + 2 (x) (2) + {(2)}^{2} \\ = x^{2} + 4 x + 4 \end{matrix}$
$\begin{matrix} (x + 2) (x + 3) & = {(x)}^{2} + (2 + 3) (x) + (2 \times 3) \\ = x^{2} + 5 x + 6 \end{matrix}$
$\begin{matrix} (x + 1) (x - 1) & = {(x)}^{2} - {(1)}^{2} \\ = x^{2} - 1 \end{matrix}$
$\begin{matrix} {(x - 1)}^{2} & = {(x)}^{2} - 2 (x) (1) + {(1)}^{2} \\ = x^{2} - 2 x + 1 \end{matrix}$
$\begin{matrix} (n + 3) (n + 5) & = {(n)}^{2} + (3 + 5) (n) + [3 \times 5] \\ = n^{2} + 8 n + 15 \end{matrix}$
$\begin{matrix} (m - 3) (m + 3) & = {(m)}^{2} - {(3)}^{2} \\ = m^{2} - 9 \end{matrix}$
$\begin{matrix} (a + b + 1) (a + b - 1) & = [(a + b) + 1] [(a + b) - 1] \\ = {(a + b)}^{2} - {(1)}^{2} \\ = a^{2} + 2 a b + b^{2} - 1 \end{matrix}$
$\begin{matrix} {(1 + b)}^{3} & = {(1)}^{3} + 3 {(1)}^{2} (b) + 3 (1) {(b)}^{2} + {(b)}^{3} \\ = 1 + 3 b + 3 b^{2} + b^{3} \end{matrix}$
$\begin{matrix} (a^{2} + 4) (a^{2} - 4) & = {(a^{2})}^{2} - {(4)}^{2} \\ = a^{4} - 16 \end{matrix}$
$\begin{matrix} {(3 a b - 5 x^{2})}^{2} & = {(3 a b)}^{2} - 2 (3 a b) (5 x^{2}) + {(5 x^{2})}^{2} \\ = 9 a^{2} b^{2} - 30 a b x^{2} + 25 x^{4} \end{matrix}$
$\begin{matrix} (a b + 3) (3 - a b) & = (3 + a b) (3 - a b) \\ = {(3)}^{2} - {(a b)}^{2} \\ = 9 - a^{2} b^{2} \end{matrix}$
$\begin{matrix} {(1 - 4 a x)}^{2} & = {(1)}^{2} + 2 (1) (4 a x) + {(4 a x)}^{2} \\ = 1 - 8 a x + 16 a^{2} x^{2} \end{matrix}$
$\begin{matrix} (a^{2} + 8) (a^{2} - 7) & = {(a^{2})}^{2} + (8 - 7) (a^{2}) + [8 (- 7)] \\ = a^{4} + a^{2} - 56 \end{matrix}$
$\begin{matrix} (x + y + 1) (x - y - 1) & = [x + (y + 1)] [x - (y + 1)] \\ = {(x)}^{2} - {(y + 1)}^{2} \\ = x^{2} - (y^{2} + 2 y + 1) \\ = x^{2} - y^{2} - 2 y - 1 \end{matrix}$
$\begin{matrix} (1 - a) (a + 1) & = (1 - a) (1 + a) \\ = 1 - a^{2} \end{matrix}$
$\begin{matrix} (m - 8) (m + 12) & = {(m)}^{2} + (- 8 + 12) (m) + [(- 8) \times 12] \\ = m^{2} + 4 m - 96 \end{matrix}$
$\begin{matrix} (x^{2} - 1) (x^{2} + 3) & = {(x^{2})}^{2} + (- 1 + 3) (x^{2}) + [(- 1) \times 3] \\ = x^{4} + 2 x^{2} - 3 \end{matrix}$
$\begin{matrix} (x^{3} + 6) (x^{3} - 8) & = {(x^{3})}^{2} + (6 - 8) (x^{3}) + [6 (- 8)] \\ = x^{6} - 2 x^{3} - 48 \end{matrix}$
$\begin{matrix} {(5 x^{3} + 6 m^{4})}^{2} & = {(5 x^{3})}^{2} + 2 (5 x^{3}) (6 m^{4}) + {(6 m^{4})}^{2} \\ = 25 x^{6} + 60 x^{3} m^{4} + 36 m^{8} \end{matrix}$
$\begin{matrix} (x^{4} - 2) (x^{4} + 5) & = {(x^{4})}^{2} + (- 2 + 5) (x^{4}) + [(- 2) \times 5] \\ = x^{8} + 3 x^{4} - 10 \end{matrix}$
$\begin{matrix} (1 - a + b) (b - a - 1) & = [(b - a) + 1] [(b - a) - 1] \\ = {(b - a)}^{2} - {(1)}^{2} \\ = b^{2} - 2 a b + a^{2} - 1 \end{matrix}$
$\begin{matrix} (a^{x} + b^{n}) (a^{x} - b^{n}) & = {(a^{x})}^{2} - {(b^{n})}^{2} \\ = a^{2 x} - b^{2 n} \end{matrix}$
$\begin{matrix} (x^{a + 1} - 8) (x^{a + 1} + 9) & = {(x^{a + 1})}^{2} + (- 8 + 9) (x^{a + 1}) + [(- 8) \times 9] \\ = x^{2 a + 2} + x^{a + 1} - 72 \end{matrix}$
$\begin{matrix} (a^{2} b^{2} + c^{2}) (a^{2} b^{2} - c^{2}) & = {(a^{2} b^{2})}^{2} - {(c^{2})}^{2} \\ = a^{4} b^{2} - c^{4} \end{matrix}$
$\begin{matrix} {(2 a + x)}^{3} & = {(2 a)}^{3} + 3 {(2 a)}^{2} (x) + 3 (2 a) {(x)}^{2} + {(x)}^{3} \\ = 8 a^{3} + 12 a^{2} x + 6 a x^{2} + x^{3} \end{matrix}$
$\begin{matrix} (x^{2} - 11) (x^{2} - 2) & = {(x^{2})}^{2} + (- 11 - 2) (x^{2}) + [(- 11) (- 2)] \\ = x^{4} - 13 x^{2} + 22 \end{matrix}$
$\begin{matrix} {(2 a^{3} - 5 b^{4})}^{2} & = {(2 a^{3})}^{2} + 2 (2 a^{3}) (5 b^{4}) + {(5 b^{4})}^{2} \\ = 4 a^{6} - 20 a^{3} b^{4} + 25 b^{8} \end{matrix}$
$\begin{matrix} (a^{3} + 12) (a^{3} - 15) & = {(a^{3})}^{2} + (12 - 15) (a^{3}) + [12 (- 15)] \\ = a^{6} - 3 a^{3} - 180 \end{matrix}$
$\begin{matrix} (m^{2} - m + n) (n + m + m^{2}) & = [(m^{2} + n) - m] [(m^{2} + n) + m] \\ = {(m^{2} + n)}^{2} - {(m)}^{2} \\ = m^{4} + 2 m n + n^{2} - m^{2} \end{matrix}$
$\begin{matrix} (x^{4} + 7) (x^{4} - 11) & = {(x^{4})}^{2} + (7 - 11) (x^{4}) + [7 (- 11)] \\ = x^{8} - 4 x^{4} - 77 \end{matrix}$
$\begin{matrix} {(11 - a b)}^{2} & = {(11)}^{2} + 2 (11) (a b) + {(a b)}^{2} \\ = 121 - 22 a b + a^{2} b^{2} \end{matrix}$
$\begin{matrix} (x^{2} y^{3} - 8) (x^{2} y^{3} + 6) & = {(x^{2} y^{3})}^{2} + (- 8 + 6) (x^{2} y^{3}) + [(- 8) \times 6] \\ = x^{4} y^{6} - 2 x^{2} y^{3} - 48 \end{matrix}$
$\begin{matrix} (a + b) (a - b) (a^{2} - b^{2}) & = (a^{2} - b^{2}) (a^{2} - b^{2}) \\ = {(a^{2} - b^{2})}^{2} \\ = {(a^{2})}^{2} - 2 (a^{2}) (b^{2}) + {(b^{2})}^{2} \\ = a^{4} - 2 a^{2} b^{2} + b^{4} \end{matrix}$
$\begin{matrix} (x + 1) (x - 1) (x^{2} - 2) & = (x^{2} - 1) (x^{2} - 2) \\ = {(x^{2})}^{2} + (- 1 - 2) (x^{2}) + [(- 1) (- 2)] \\ = x^{4} - 3 x^{2} + 2 \end{matrix}$
$\begin{matrix} (a + 3) (a^{2} + 9) (a - 3) & = (a^{2} - 9) (a^{2} + 9) \\ = {(a^{2})}^{2} - {(9)}^{2} \\ = a^{4} - 81 \end{matrix}$
$\begin{matrix} (x + 5) (x - 5) (x^{2} + 1) & = (x^{2} - 25) (x^{2} + 1) \\ = {(x^{2})}^{2} + (- 25 + 1) (x^{2}) + [(- 25) \times 1] \\ = x^{4} - 24 x - 25 \end{matrix}$
$\begin{matrix} (a + 1) (a - 1) (a + 2) (a - 2) & = (a^{2} - 1) (a^{2} - 4) \\ = {(a^{2})}^{2} + (- 1 - 4) (a^{2}) + [(- 1) (- 4)] \\ = a^{4} - 5 a^{2} + 4 \end{matrix}$
$\begin{matrix} (a + 2) (a - 3) (a - 2) (a + 3) & = (a^{2} - 4) (a^{2} - 9) \\ = {(a^{2})}^{2} + (- 4 - 9) (a^{2}) + [(- 4) (- 9)] \\ = a^{4} - 13 a^{2} + 36 \end{matrix}$