Double Math

# (a+b)^3 formula

This \left(a+b\right)^3 formula is used to find the cube of the polynomial.This \left(a+b\right)^3 formula is an algebraic identity.

Formula

\left(a+b\right)^3=a^3+3ab(a+b)+b^3

### How to derive Formula of (a+b)^3 ?

We multiply (a+b) three times just like (a+b)(a+b)(a+b). This formula is also an identity.

\left(a+b\right)^3 = (a+b)(a+b)(a+b)

= (a+b)(a^2+2ab+b^2)\\

= a(a^2+2ab+b^2)+b(a^2+2ab+b^2)\\

= a^3+2a^2b+ab^2+a^2b+2ab^2+b^3\\

= a^3+3a^2b+3ab^2+b^3\\

= a^3+3ab(a+b)+b^3\\

Solve Problems by using \left(a+b\right)^3 formula:

Example#1:

={(2x+3y)}^3\;.

={(2x)}^3+3(2x)(3y)(2x+3y)+{(3y)}^3\;.

=8x^3+18xy(2x+3y)+27y^3\;.

=8x^3+36x^2y+54xy^2+27y^3\;.

Example # 2:

={(5x+2y)}^3\;.

={(5x)}^3+3(5x)(2y)(5x+2y)+{(2y)}^3\;.

=125x^3+30xy(5x+2y)+8y^3\;.

=125x^3+150x^2y+60xy^2+8y^3\;.

Find unknown values by using (a+b)^3

Find the value when ab=1 , a+2b=5 , a^3+8b^3\;.

{(a+2b)}^3=a^3+3(a)(2b)(a+2b)+{(2b)}^3\;.

{(5)}^3=a^3+6ab(5)+8b^3\;.

125=a^3+30ab+8b^3\;.

125=a^3+30(1)+8b^3\;.

125-30=a^3+8b^3\;.

95=a^3+8b^3\;.