(a+b)^3 formula

Leave a Comment / Mathematics / By Azhar

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 , $a^3+8b^3\;$ .

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

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

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

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

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

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

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