Chapter 11.7, Problem 19WE

To determine

To calculate:

The expansion and simplification of

  ${\left(a^2+2ab+b^2\right)}^3$

Answer to Problem 19WE

The simplified expression is $a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6$ .

Explanation of Solution

Given information:

The given expression is,

  ${\left(a^2+2ab+b^2\right)}^3$

Formula:

The formula used is,

  ${\left(a^2+2ab+b^2\right)}^3=\left(a^2+2ab+b^2\right)\left(a^2+2ab+b^2\right)\left(a^2+2ab+b^2\right)$

Calculation:

The given expression is,

  ${\left(a^2+2ab+b^2\right)}^3$

In the first step perform, the multiplication of $\left(a^2+2ab+b^2\right)\left(a^2+2ab+b^2\right)$ ,

  $\left(a^2+2ab+b^2\right)\times \left(a^2+2ab+b^2\right)=\left(a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4\right)$

Now, in the next step

  $\begin{array}{l}=\left(a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4\right)\times \left(a^2+2ab+b^2\right)\\ =a^4{a}^2+a^4\times 2ab+a^4{b}^2+4a^{3}b{a}^2+\left[\therefore \text{Distributive}\text{property}\right]\\ 4a^{3}b\times 2ab+4a^{3}b{b}^2+6a^2{b}^2{a}^2+6a^2{b}^2\times 2ab+6a^2\\ =a^4{a}^2+2a^{4}ab+a^4{b}^2+4a^{3}b{a}^2+4\times 2a^{3}abb+4a^3{b}^{2}b+6a^2{a}^2{b}^2+6\times 2a^{2}a{b}^{2}b+6a^2{b}^2{b}^2+4a^{2}a{b}^3+4\end{array}$

Simplifying the obtained expression,

  $\begin{array}{l}=a^4{a}^2+2a^{4}ab+a^4{b}^2+4a^{3}b{a}^2+4\times 2a^{3}abb+4a^3{b}^{2}b+\\ 6a^2{a}^2{b}^2+6\times 2a^{2}a{b}^{2}b+6a^2{b}^2{b}^2+4a^{2}a{b}^3+4\\ =a^6+2a^{5}b+a^4{b}^2+4a^{5}b+8a^4{b}^2+4a^3{b}^3+6a^4{b}^2+\\ 12a^3{b}^3+6a^2{b}^4+4a^3{b}^3+8a^2{b}^4+4a{b}^5+a^2{b}^4+2a{b}^5+b^6\end{array}$

  $=a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6\left[\because \text{Adding}\text{the}\text{similar}\text{elements}\right]$

The expression obtained is,

  $a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6$