Question

Cum se rezolva $(a+b) ^{2} (a+b) ^{2}$ si $(a-b) ^{2} (a+b) ^{2}$ ?????

Answer 1

(a+b)²(a+b)²=(a²+2ab+b²)(a²+2ab+b²) = a⁴+2a³b+a²b²+2a³b+4a²b²+2ab³+a²b²+2ab³+b⁴ = a⁴+4a³b+6a²b²+4ab³+b⁴
(a-b)²(a+b)² = (a²-2ab+b²)(a²+2ab+b²) = a⁴+2a³b+a²b²-2a³b-4a²b²-2ab³+a²b²+2ab³+b⁴=a⁴-2a²b²+b⁴

Answer 2

$(a+b)^2(a+b)^2=(a^2+2ab+b^2)(a^2+2ab+b^2)= \\ =a^2 \cdot a^2+a^2 \cdot 2ab+a^2 \cdot b^2+2ab \cdot a^2+2ab \cdot 2ab+2ab \cdot b^2+b^2 \cdot a^2+ \\ +b^2 \cdot 2ab+b^2 \cdot b^2= \\ =a^4+2a^3b+a^2b^2+2a^3b+4a^2b^2+2ab^3+a^2b^2+2ab^3+b^4= \\ =\boxed{a^4+4a^3b+6a^2b^2+4ab^3+b^4}$

$\displaystyle (a-b)^2(a+b)^2=(a^2-2ab+b^2)(a^2+2ab+b^2)= \\ = a^2 \cdot a^2+a^2 \cdot 2ab+a^2 \cdot b^2-2ab \cdot a^2-2ab \cdot 2ab-2ab \cdot b^2+b^2 \cdot a^2+ \\ +b^2 \cdot 2ab+b^2 \cdot b^2= \\ =a^4+2a^3b+a^2b^2 -2a^3b-4a^2b^2-2ab^3+a^2b^2+2ab^3+b^4= \\ =\boxed{a^4-2a^2b^2+b^4}$