Chapter 1.4, Problem 35E

To determine

To calculate: Thesimplified form of the expression $\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}$ .

Answer to Problem 35E

The simplified form of the expression is $x^3+x^2$ .

Explanation of Solution

Given information:

The expression is $\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}$ .

Formula used:

If a, b, c and d are any numbers then $\frac{\frac{a}{b}}{\frac{c}{d}}$ is expressed by $\frac{a\times b}{c\times d}$ .

The sum of the square of two numbers is algebraically expressed as ${\left(a+b\right)}^2=a^2+b^2+2ab$ .

Calculation:

Consider the provided expression $\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}$ .

Recall that if a, b, c and d are any numbers then $\frac{\frac{a}{b}}{\frac{c}{d}}$ is expressed by $\frac{a\times b}{c\times d}$ . Apply it in the above expression.

  $\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}=\frac{x^3\times \left(x^2+2x+1\right)}{\left(x+1\right)\times x}$

Recall that the sum of the square of two numbers is algebraically expressed as ${\left(a+b\right)}^2=a^2+b^2+2ab$ . Apply it,

  $\begin{array}{c}\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}=\frac{x^3\times \left(x^2+2x+1\right)}{\left(x+1\right)\times x}\\ =\frac{x\cdot x^2}{x}\cdot \frac{\left(\left(x^2+2\times x\times 1+1^2\right)\right)}{\left(x+1\right)}\\ =\frac{x^2\times {\left(x+1\right)}^2}{\left(x+1\right)}\end{array}$

Strike off the common terms,

  $\begin{array}{c}\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}=\frac{x^2\times {\left(x+1\right)}^2}{\left(x+1\right)}\\ =\frac{x^2\times \left(x+1\right)\left(x+1\right)}{\left(x+1\right)}\\ =x^2\left(x+1\right)\end{array}$

Expand the expression obtained above,

  $\begin{array}{c}\frac{\frac{x^3}{x+1}}{\frac{x}{x^2+2x+1}}=\frac{x^2\times {\left(x+1\right)}^2}{\left(x+1\right)}\\ =\frac{x^2\times \left(x+1\right)\left(x+1\right)}{\left(x+1\right)}\\ =x^2\left(x+1\right)\\ =x^3+x^2\end{array}$

Thus, the simplified form of the expression is $x^3+x^2$ .