Question

Докажите тождество. 2(а^2+аb+b^2)^2 = а^4+b^4+(a+b)^4

Answer 1

$2(a^2+ab+b^2)^2=a^4+b^4+(a+b)^4\\\\1)\; \; (a^2+ab+b^2)^2=((a^2+b^2)+ab)^2=\\\\=(a^2+b^2)^2+2ab(a^2+b^2)+a^2b^2=\\\\=a^4+2a^2b^2+b^4+2a^3b+2ab^3+a^2b^2=\\\\=a^4+2a^3b+3a^2b^2+2ab^3+b^4$

$2)\; \; (a+b )^4=((a+b)^2)^2=(a^2+2ab+b^2)^2=\\\\=(a^2+2ab+b^2)\cdot (a^2+2ab+b^2)=\\\\=a^4+2a^3b+a^2b^2+2a^3b+4a^2b^2+2ab^3+a^2b^2+2ab^3+b^4=\\\\=a^4+4a^3b+6a^2b^2+4ab^3+b^4$

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