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CAPITULO VI

PRODUCTOS Y COCIENTES NOTABLES

Ejercicio 71

Hallar, por simple inspección, el cociente de:

$\frac{x^{4} - y^{4}}{x - y} = x^{3} + x^{2} y + x y^{2} + y^{3}$
$\frac{m^{5} + n^{5}}{m + n} = m^{4} - m^{3} n + m^{2} n^{2} - m n^{3} + n^{4}$
$\frac{a^{5} - n^{5}}{a - n} = a^{4} + a^{3} n + a^{2} n^{2} + a n^{3} + n^{4}$
$\frac{x^{6} - y^{6}}{x + y} = x^{5} - x^{4} y + x^{3} y^{2} - x^{2} y^{3} + x y^{4} - y^{5}$
$\frac{a^{6} - b^{6}}{a - b} = a^{5} + a^{4} b + a^{3} b^{2} + a^{2} b^{3} + a b^{4} + b^{5}$
$\frac{x^{7} + y^{7}}{x + y} = x^{6} - x^{5} y + x^{4} y^{2} - x^{3} y^{3} + x^{2} y^{4} - x y^{5} + y^{6}$
$\frac{a^{7} - m^{7}}{a - m} = a^{6} + a m^{5} + a^{2} m^{4} + a^{3} m^{3} + a^{2} m^{4} + a m^{5} + m^{6}$
$\frac{a^{8} - b^{8}}{a + b} = a^{7} - a^{6} b + a^{5} b^{2} - a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5} + a b^{6} - b^{7}$
$\frac{x^{10} - y^{10}}{x - y} = x^{9} + x^{8} y + x^{7} y^{2} + x^{6} y^{3} + x^{5} y^{4} + x^{4} y^{5} + x^{3} y^{6} + x^{2} y^{7} + x y^{8} + y^{9}$
$\frac{m^{9} + n^{9}}{m + n} = m^{8} - m^{7} n + m^{6} n^{2} - m^{5} n^{3} + m^{4} n^{4} - m^{3} n^{5} + m^{2} n^{6} - m n^{7} + n^{8}$
$\frac{m^{9} - n^{9}}{m - n} = m^{8} + m^{7} n + m^{6} n^{2} + m^{5} n^{3} + m^{4} n^{4} + m^{3} n^{5} + m^{2} n^{6} + m n^{7} + n^{8}$
$\frac{a^{10} - x^{10}}{a + x} = a^{9} - a^{8} x + a^{7} x^{2} - a^{6} x^{3} + a^{5} x^{4} - a^{4} x^{5} + a^{3} x^{6} - a^{2} x^{7} + a x^{8} - x^{9}$
$\frac{1 - n^{5}}{1 - n} = 1 + n + n^{2} + n^{3} + n^{4}$
$\frac{1 - a^{6}}{1 - a} = 1 + a + a^{2} + a^{3} + a^{5}$
$\frac{1 + a^{7}}{1 + a} = 1 - a + a^{2} - a^{3} + a^{4} - a^{5} + a^{6}$
$\frac{1 - m^{8}}{1 + m} = 1 - m + m^{2} - m^{3} + m^{4} - m^{5} + m^{6} - m^{7}$
$\frac{x^{4} - 16}{x - 2} = x^{3} + 2 x^{2} + 4 x + 8$
$\frac{x^{6} - 64}{x + 2} = x^{5} - 2 x^{4} + 4 x^{3} - 8 x^{2} + 16 x - 32$
$\frac{x^{7} - 128}{x - 2} = x^{6} + 2 x^{5} + 4 x^{4} + 8 x^{3} + 16 x^{2} + 32 x + 64$
$\frac{a^{5} + 243}{a + 3} = a^{4} - 3 a^{3} + 9 a^{2} - 27 a + 81$
$\frac{x^{6} - 729}{x - 3} = x^{5} + 3 x^{4} + 9 x^{3} + 27 x^{2} + 81 x + 243$
$\frac{625 - x^{4}}{x + 5} = 125 - 25 x + 5 x^{2} - x^{3}$
$\frac{m^{8} - 256}{m - 2} = m^{7} + 2 m^{6} + 4 m^{5} + 8 m^{4} + 16 m^{3} + 32 m^{2} + 64 m + 128$
$\frac{x^{10} - 1}{x - 1} = x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1$
$\frac{x^{5} + 243 y^{5}}{x + 3 y} = x^{4} - 3 x^{3} y + 9 x^{2} y^{2} - 27 x y^{3} + 81 y^{4}$
$\frac{16 a^{4} - 81 b^{4}}{2 a - 3 b} = 8 a^{3} + 12 a^{2} b + 18 a b^{2} + 27 b^{3}$
$\frac{64 m^{6} - 729 n^{6}}{2 m + 3 n} = 32 m^{5} - 48 m^{4} n + 72 m^{3} n^{2} - 108 m^{2} n^{3} + 162 m n^{4} - 243 n^{5}$
$\frac{1024 x^{10} - 1}{2 x - 1} = 512 x^{9} + 256 x^{8} + 128 x^{7} + 64 x^{6} + 32 x^{5} + 16 x^{4} + 8 x^{3} + 4 x^{2} + 2 x + 1$
$\frac{512 a^{9} + b^{9}}{2 a + b} = 256 a^{8} - 128 a^{7} b + 64 a^{6} b^{2} - 32 a^{5} b^{3} + 16 a^{4} b^{4} - 8 a^{3} b^{5} + 4 a^{2} b^{6} - 2 a b^{7} + b^{8}$
$\frac{a^{6} - 729}{a - 3} = a^{5} + 3 a^{4} + 9 a^{3} + 27 a^{2} + 81 a + 243$