Question

A expressão : a^4 -b^4+2a^3b-2ab^3 é igual a: a)(a-b)^3. (a+b) b)(a^2+b^2)(a-b)^2 c)(a+b)^2 (a-b)^2 d)(a+b)^2 (a-b)^3 e)(a+b)^3 (a-b)

Answer 1

$a^4 -b^4+2a^3b-2ab^3= a^4 +2 a^3 b -b^4 -2 a b^3 =a^3 (a+2b)-b^3(b+2a) \\ = a^3([a + b] + b)-b^3 ([a+b]+a)=a^3 (a+b)+a^3 b -b^3 (a+b) - b^3 a \\ = (a^3 -b^3 )(a+b)+a ^3 b- b^3 a = (a^3 -b^3 )(a+b)+(ab)(a ^2- b^2)$

Sabemos que $(a^3 - b^3) = (a-b)(a^2 +ab + b^2)$. Portanto,

$(a^3 -b^3 )(a+b)+(ab)(a ^2- b^2) \\ =(a-b)(a^2 +ab + b^2)(a+b)+(ab)(a^2-b^2) \\ = (a^2 -b^2)(a^2 + ab+ b^2)+ (ab)(a^2 - b^2) \\ = (a^2 -b^2)(a^2 + ab + b^2 + ab) \\ = (a^2 - b^2)(a^2 +2ab + b^2 ) \\ = (a^2 - b^2 )(a+b)^2 \\=(a-b)(a+b)(a+b)^2 \\=(a-b)(a+b)^3$