1. ## algebra

Solve
Show_that
(a-b)^2(a+b)+ab(a+b)=a^3+b^3

2. Originally Posted by Crystal09
Solve
Show_that
(a-b)^2(a+b)+ab(a+b)=a^3+b^3
start with the left side. expand everything and cancel what should be canceled. then show you can get the left side. hopefully you didn't make a mistake with the signs, i haven't checked it yet

3. Hello, Crystal09!

Show that: .$\displaystyle (a-b)^2(a+b)+ab(a+b)\:=\:a^3+b^3$
What's stopping you from multiply it out?

If you looking for a deliberately complicated proof . . .

We have: .$\displaystyle a^3 + b^3$

Factor: .$\displaystyle (a + b)(a^2-ab+b^2)$

Subtract and add $\displaystyle ab\!:\;\;(a + b)(a^2 \:{\color{red}-\: ab}\: - ab + b^2 \:{\color{red}+\: ab}\,)$

And we have: .$\displaystyle (a + b)(a^2 - 2ab + b^2 + ab)$

Factor: .$\displaystyle (a+b)\left([a-b]^2 + ab\right)$

Distribute: .$\displaystyle (a+b)(a-b)^2 +(a+b)ab \;=\;(a-b)^2(a+b) + ab(a+b)$

4. Yeah, what's stopping you from multiply it out?

I can try to solve the half of the algebraic expression you're struggling with, in order to help you solving the remaining under;

$\displaystyle (a-b)^2 \cdot (a+b) + ab(a+b) = a^3 + b^3$
$\displaystyle (a^2-2ab+b^2) \cdot (a+b) + (a^2b + ab^2) = a^3 + b^3$