Question

cada um dos polinômios seguintes representa uma diferença de dois quadrados.fatore esses polinômios.

a) a⁴-c⁴


b) a⁶b⁶-0,01


c) x⁴-100



d) 9-y⁶



e) 81r²-S⁴​

Answer 1

a)

$= a {}^{4} - c {}^{4} \\ = a {}^{2 \: . \: 2} - c {}^{2 \: . \: 2} \\ (a {}^{2} ) {}^{2} - (c {}^{2} ) {}^{2} \\ = (a {}^{2} - c {}^{2} ) \: . \: (a {}^{2} + c {}^{2} ) \\ \boxed{ = (a - c) \: . \: (a + c) \: . \: (a {}^{2} + c {}^{2}) }$

b)

$= a {}^{6} b {}^{6} - 0,01 \\ a {}^{6} b {}^{6} - \frac{1}{100} \\ = a {}^{2 \: . \: 3} b {}^{2 \: . \: 3} - \frac{1 {}^{2} }{10 {}^{2} } \\ = (a {}^{3} ) {}^{2} \: . \: (b {}^{3} ) {}^{2} - ( \frac{1}{10} ) {}^{2} \\ = (a {}^{3} b {}^{3} ) {}^{2} - ( \frac{1}{10} ) {}^{2} \\ \boxed{ = (a {}^{3} b {}^{3} - \frac{1}{10} ) \: . \: (a {}^{3}b {}^{3} + \frac{1}{10}) }$

c)

$= x {}^{4} - 100 \\ = x {}^{2 \: . \: 2} - 10 {}^{2} \\ = (x {}^{2} ) {}^{2} - 10 {}^{2} \\ \boxed{ = (x {}^{2} - 10) \: . \: (x {}^{2} + 10)}$

d)

$= 9 - y {}^{6} \\ = 3 {}^{2} - y {}^{2 \: . \: 3} \\ = 3 {}^{2} - (y {}^{3} ) {}^{2} \\ \boxed{ = (3 - y {}^{3}) \: . \: (3 + y {}^{3} ) }$

e)

$= 81r {}^{2} - s {}^{4} \\ = 9 {}^{2} r {}^{2} - s {}^{2 \: . \: 2} \\ = (9r) {}^{2} - (s {}^{2} ) {}^{2} \\ \boxed{ = (9r - s {}^{2}) \: . \: (9r + s {}^{2} )}$

Att. NLE Top Shotta