Question

Resolva as expressões, simplificando-as no final
(obs: O circunflexo ( ^ ) antes do número, significa potenciação.

1) (m+n)(m-n) - (m-n)²

2) (x+2)(x-2)(x²+4)(x^4+16)

3) (a+b)² - (a+b)(a-b) - 2ab + 2(a-b)² - 4ab²

4) (m+n)(m²-mn+n²) - (m-n)(m²+mn+n²)

Answer 1

$1)\;\;(m+n)(m-n)-(m-n)^2=(m^2-n^2) - (m^2 - 2mn +n^2) =\\ -2n(m+n)$

$2)\;\;(x+2)(x-2)(x^2+4)(x^4+16)=(x^2-2^2)(x^2+2^2)(x^4+2^4)=\\ (x^4-2^4)(x^4+2^4) = (x^8-2^8) = (x^8-256)$

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

$4)\;\;(m+n)(m^2-mn+n^2) - (m-n)(m^2+mn+n^2) =\\ (m+n)^3-(m-n)^3\\\\ chamemos\; (m+n)=a\; e\; (m-n) = b\\\\ a^3-b^3 = (a-b)(a^2+ab+b^2);\\\\ Substituindo\; os\; valores\; agora:\\\\ ((m+n)-(m-n))((m+n)^2 + (m+n)(m-n)+(m-n)^2)=\\ (2n)((m^2+2mn+n^2)+(m^2-n^2)+(m^2-2mn+n^2)=\\ (2n)(3m^2+n^2)$

Abraços,
Haller