Chapter 8.4, Problem 58HP

a.

To determine

To find the product of $\left(a+b\right)\left(a+b\right)\left(a+b\right)$ .

Answer to Problem 58HP

  $a^3+3a^{2}b+3a{b}^2+b^3$

Explanation of Solution

Given:

  ${(a+b)}^3$

Calculation:

Consider the given expression,

  $\Rightarrow {(a+b)}^3$

At first splitting the expression,

  $\begin{array}{l}$ \Rightarrow {(a+b)}^3$\Rightarrow \left(a+b\right)\left(a+b\right)\left(a+b\right)\end{array}$

Now, using distributive property,

  $\begin{array}{l}\Rightarrow (a+b)\left[a(a+b)+b(a+b)\right]\\ \Rightarrow (a+b)[a^2+2ab+b^2]\end{array}$

  $\begin{array}{l}\Rightarrow a[a^2+2ab+b^2]+b[a^2+2ab+b^2]\\ \Rightarrow a^3+2a^{2}b+a{b}^2+a^{2}b+2a{b}^2+b^3\\ \Rightarrow a^3+3a^{2}b+3a{b}^2+b^3\end{array}$

Conclusion:

Therefore, the product of $\left(a+b\right)\left(a+b\right)\left(a+b\right)$ is $a^3+3a^{2}b+3a{b}^2+b^3$ .

b.

To determine

To find the expression for ${(x+2)}^3$ .

Answer to Problem 58HP

  $x^3+6x^2+12x+8$

Explanation of Solution

Given:

  ${(x+2)}^3$

Calculation:

Consider the given expression,

  $\Rightarrow {(x+2)}^3$

At first splitting the expression,

  $\begin{array}{l}\Rightarrow {(}^x\\ \Rightarrow \left(x+2\right)\left(x+2\right)\left(x+2\right)\end{array}$

Now, using distributive property,

  $\begin{array}{l}\Rightarrow (x+2)\left[x(x+2)+2(x+2)\right]\\ \Rightarrow (x+2)[x^2+2\times x+2\times x+2^2]\\ \Rightarrow (x+2)[x^2+4x+4]\end{array}$

  $\begin{array}{l}\Rightarrow x[x^2+4x+4]+2[x^2+4x+4]\\ \Rightarrow x^3+2\times x^2\times 2+x\times 2^2+x^2\times 2+2x\times 2^2+b^3\\ \Rightarrow x^3+6x^2+12x+8\end{array}$

Conclusion:

Therefore, the pattern expression is $x^3+6x^2+12x+8$ .

c.

To determine

Diagram for the geometric model of ${(a+b)}^3$ .

Explanation of Solution

Given:

  ${(a+b)}^3$

Formula used:

  ${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

Calculation:

Consider the given expression,

  $\Rightarrow {(a+b)}^3$

It can be written as,

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

So, geometric model is

  

Conclusion:

Therefore, the above diagram represents ${(a+b)}^3.$

d.

To determine

Pattern of the cube of difference ${(a-b)}^3$ .

Answer to Problem 58HP

  $a^3-b^3-3ab(a-b)$

Explanation of Solution

Given:

  ${(a-b)}^3$

Calculation:

Consider the given expression,

  $\Rightarrow {(a-b)}^3$

At first splitting the expression,

  $\begin{array}{l}$ \Rightarrow {(a-b)}^3$\Rightarrow \left(a-b\right)\left(a-b\right)\left(a-b\right)\end{array}$

Now, using distributive property,

  $\begin{array}{l}\Rightarrow (a-b)\left[a(a-b)-b(a-b)\right]\\ \Rightarrow (a-b)[a^2-2ab+b^2]\end{array}$

  $\begin{array}{l}\Rightarrow a[a^2-2ab+b^2]-b[a^2-2ab+b^2]\\ \Rightarrow a^3-2a^{2}b+a{b}^2-a^{2}b+2a{b}^2-b^3\\ \Rightarrow a^3-3a^{2}b+3a{b}^2+b^3\\ \Rightarrow a^3-b^3-3ab(a-b)\end{array}$

Conclusion:

Therefore, pattern of the cube of difference is $a^3-b^3-3ab(a-b)$ .