Question

Factorize: $a^{12} + b^{12}$ Please provide full explanation.

Answer 1

Answer:

(a + b) (a-b) (a^2 + b^2) (a^2 + b^2 - ab) (a^2 + b^2 + ab) ( a^4 + b^4 - a^2 b^2)

Step-by-step explanation:

a^12 - b^12

= ( a^6)^2 - (b^6)^2

= ( a^6 - b^6) (a^6 + b^6)

= [ (a^2)^3 - (b^2)^3] [ (a^2)^3 + (b^2)^3]

= (a^2 - b^2) (a^4 + b^4 + a^2 b^2) (a^2 + b^2) (a^4 + b^4 - a^2 b^2)

= (a + b) (a - b) ( a^2 +b^2) (a^4 + b^4 + 2a^2 b^2 - a^2 b^2) ( a^4 + b^4 - a^2 b^2)

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

= (a + b) (a-b) (a^2 + b^2) (a^2 + b^2 - ab) (a^2 + b^2 + ab) ( a^4 + b^4 - a^2 b^2)

Hope it helps you