Factorize: (a + b)^3 – 8(a – b)^3.

Question

this is very urgent

Answer 1

Let a+b=A, and 2(a-b)=B, so the given equation can be restated:

(a+b)³-8(a-b)³=A³-B³ ··· Eq.1    Use the standard polynomial form for A³-B³.   We have:

A³-B³=(A-B)(A²+AB+B²) ··· Eq.2    Plug A=a+b, and B=2(a-b) into each term of LHS.   We have:

A-B=(a+b)-2(a-b)=3b-a

A²=(a+b)²=a²+2ab+b²

AB=(a+b)x2(a-b)=2a²-2b²

B²=2²(a-b)²=4a²-8ab+4b²    Thus, we have:

A²+AB+B²=7a²-6ab+3b²    From Eq.1 and Eq.2, we have:

(a+b)³-8(a-b)³=(3b-a)(7a²-6ab+3b²) ··· Eq.3

CK: Plug a=2, b=1 (or any number) into Eq.3   LHS=(2+1)³-8(2-1)³=27-8=19   While,

RHS=(3(1)-2)(7(2)²-6(2)(1)+3(1)²)=(1)(28-12+3)=19   Thus, LHS=RHS    CKD.

The answer: (a+b)³-8(a-b)³=(3b-a)(7a²-6ab+3b²)