Question

Demonstrați identitățile:
a. a^4-b^4=(a-b)(a+b)(a^2-b^2)
b. a^5-b^5=(a-b)(a^4+a^3×b+ a^2×b^2+a×b^3+b^4)

Answer 1

$a^{n} -b^{n}=(a-b)[(a^{n-1}b+a^{n-2}\cdot b^{2}b^{3}+.....+a\cdot b^{n-1}]$  Formula binomului general

$a) a^{4}-b^{4}=(a^{2} -b^{2})(a^{2} +b^{2}) =(a-b)(a+b)(a^{2} +b^{2} )\\ \\b)a^{5}-b^{5}=\\ \\ a^{n} -b^{n}=(a-b)[(a^{n-1}b+a^{n-2}\cdot b^{2}b^{3}+.....+a\cdotb^{n-1}]\\ \\ a^{5}-b^{5}=(a-b)(a^{4}+a^{3}\cdot b+a^{2}\cdot b^{2} +a\cdot b^{3}+b^{4})$