Question

Simplify the rational expression.

$\frac{2x^3-x^2-6x}{2x^2-7x+6}$

Short answer

The simplified rational expression is$\frac{x\left(2x+3\right)}{2x-3}$

Step-by-step solution

Step 1. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Step 2. Analyze the expression.

Given$\frac{2x^3-x^2-6x}{2x^2-7x+6}$

This is a rational expression.

To simplify the rational expression, we factor both numerator and denominator and use this$\frac{AC}{BC}=\frac{A}{B}$

Therefore, to simplify a rational expression, we cancel factors that are common to the numerator and denominator.

Step 3. Simplify the expression.

Consider

$\begin{array}{l}\frac{2x^3-x^2-6x}{2x^2-7x+6}=\frac{x\left(2x^2-x-6\right)}{2x^2-4x-3x+6}\\ \Rightarrow \frac{2x^3-x^2-6x}{2x^2-7x+6}=\frac{x\left(2x^2-4x+3x-6\right)}{2x\left(x-2\right)-3\left(x-2\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\\ \Rightarrow \frac{2x^3-x^2-6x}{2x^2-7x+6}=\frac{x\left(2x\left(x-2\right)+3\left(x-2\right)\right)}{2x\left(x-2\right)-3\left(x-2\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\\ \Rightarrow \frac{2x^3-x^2-6x}{2x^2-7x+6}=\frac{x\left(2x+3\right)\left(x-2\right)}{\left(2x-3\right)\left(x-2\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\end{array}$

The common factor to the numerator and denominator is$x-2$

So we will cancel the common factor from the numerator and denominator.

The simplified expression is$\frac{x\left(2x+3\right)\left(x-2\right)}{\left(2x-3\right)\left(x-2\right)}=\frac{x\left(2x+3\right)}{2x-3}$