Chapter 6.2, Problem 41WE

To determine

To calculate: The simplified form of the expression $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}$.

Answer to Problem 41WE

The simplified form of the expression $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}$ is $\frac{5x^3\left(x+1\right)}{12}$.

Explanation of Solution

Given information:

The expression $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}$.

Formula used:

Fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

To factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

Calculation:

Consider the expression $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}$.

Take the trinomial $x^2-x-2$.

Recall that to factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

For the above polynomial $-2$ and $1$ are two numbers such that their sum is $-1$ and product is $-2$.

Therefore, the trinomial is rewritten as,

  $x^2-x-2=x^2-2x+x-2$

Take out the common terms,

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

Take out the common factor that is $6$ from the denominator of first term,

  $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}=\frac{10x^4}{6\left(x-2\right)}\cdot \frac{\left(x-2\right)\left(x+1\right)}{4x}$

Strike off the similar terms,

  $\begin{array}{c}\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}=\frac{10x^4}{6\left(x-2\right)}\cdot \frac{\left(x-2\right)\left(x+1\right)}{4x}\\ =\frac{10x^3}{6}\cdot \frac{\left(x+1\right)}{4}\end{array}$

Recall that fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

Apply it,

  $\begin{array}{c}\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}=\frac{10x^4}{6\left(x-2\right)}\cdot \frac{\left(x-2\right)\left(x+1\right)}{4x}\\ =\frac{10x^3}{6}\cdot \frac{\left(x+1\right)}{4}\\ =\frac{10x^3\left(x+1\right)}{24}\\ =\frac{5x^3\left(x+1\right)}{12}\end{array}$

Thus, the simplified form of the expression $\frac{10x^4}{6x-12}\cdot \frac{x^2-x-2}{4x}$ is $\frac{5x^3\left(x+1\right)}{12}$.