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\;.