Chapter 6.2, Problem 42WE

To determine

To calculate: The simplified form of the expression $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}$.

Answer to Problem 42WE

The simplified form of the expression $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}$ is $-9\left(x+3\right)$.

Explanation of Solution

Given information:

The expression $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}$.

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 $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}$.

Take the trinomial $2x^2+5x-3$.

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 $6$ and $-1$ are two numbers such that their sum is $5$ and product is $-6$.

Therefore, the trinomial is rewritten as,

  $2x^2+5x-3=2x^2+6x-x-3$

Take out the common terms,

  $\begin{array}{c}2x^2+5x-3=2x^2+6x-x-3\\ =2x\left(x+3\right)-1\left(x+3\right)\\ =\left(x+3\right)\left(2x-1\right)\end{array}$

Take out the common factor that is $9$ from the numerator of second term,

  $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}=\frac{\left(x+3\right)\left(2x-1\right)}{x+2}\cdot \frac{9\left(x+2\right)}{1-2x}$

Strike off the similar terms,

  $\begin{array}{c}\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}=\frac{\left(x+3\right)\left(2x-1\right)}{x+2}\cdot \frac{9\left(x+2\right)}{1-2x}\\ =-\frac{\left(x+3\right)}{1}\cdot \frac{9}{1}\end{array}$

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

Apply it,

  $\begin{array}{c}\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}=\frac{\left(x+3\right)\left(2x-1\right)}{x+2}\cdot \frac{9\left(x+2\right)}{1-2x}\\ =-\frac{\left(x+3\right)}{1}\cdot \frac{9}{1}\\ =-9\left(x+3\right)\end{array}$

Thus, the simplified form of the expression $\frac{2x^2+5x-3}{x+2}\cdot \frac{9x+18}{1-2x}$ is $-9\left(x+3\right)$.