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Have you learned the basics of rational expression simplification? Great! Now gain more experience with some trickier examples.

### What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. A rational expression is considered simplified if the numerator and denominator have no factors in common.
If this is new to you, we recommend that you check out our intro to simplifying rational expressions.

### What you will learn in this lesson

In this lesson, you will practice simplifying more complicated rational expressions. Let's look at two examples, and then you can try some problems!

## Example 1: Simplifying ‍

Step 1: Factor the numerator and denominator
Here it is important to notice that while the numerator is a monomial, we can factor this as well.
$\frac{10{x}^{3}}{2{x}^{2}-18x}=\frac{2\cdot 5\cdot x\cdot {x}^{2}}{2\cdot x\cdot \left(x-9\right)}$
Step 2: List restricted values
From the factored form, we see that $x\ne 0$ and $x\ne 9$.
Step 3: Cancel common factors
$\begin{array}{rl}\frac{2\cdot 5\cdot x\cdot {x}^{2}}{2\cdot x\cdot \left(x-9\right)}& =\frac{\overline{)2}\cdot 5\cdot \overline{)x}\cdot {x}^{2}}{\overline{)2}\cdot \overline{)x}\cdot \left(x-9\right)}\\ \\ & =\frac{5{x}^{2}}{x-9}\end{array}$
We write the simplified form as follows:
$\frac{5{x}^{2}}{x-9}$ for $x\ne 0$

### Main takeaway

In this example, we see that sometimes we will have to factor monomials in order to simplify a rational expression.

1) Simplify $\frac{6{x}^{2}}{12{x}^{4}-9{x}^{3}}$.

## Example 2: Simplifying ‍

Step 1: Factor the numerator and denominator
While it does not appear that there are any common factors, $x-3$ and $3-x$ are related. In fact, we can factor $-1$ out of the numerator to reveal a common factor of $x-3$.
$\begin{array}{rl}& \phantom{=}\frac{\left(3-x\right)\left(x-1\right)}{\left(x-3\right)\left(x+1\right)}\\ \\ & =\frac{-1\left(-3+x\right)\left(x-1\right)}{\left(x-3\right)\left(x+1\right)}\\ \\ & =\frac{-1\left(x-3\right)\left(x-1\right)}{\left(x-3\right)\left(x+1\right)}\phantom{\rule{1em}{0ex}}\text{Commutativity}\end{array}$
Step 2: List restricted values
From the factored form, we see that $x\ne 3$ and $x\ne -1$.
Step 3: Cancel common factors
$\begin{array}{rl}& \phantom{=}\frac{-1\left(x-3\right)\left(x-1\right)}{\left(x-3\right)\left(x+1\right)}\\ \\ \\ & =\frac{-1\overline{)\left(x-3\right)}\left(x-1\right)}{\overline{)\left(x-3\right)}\left(x+1\right)}\\ \\ & =\frac{-1\left(x-1\right)}{x+1}\\ \\ & =\frac{1-x}{x+1}\end{array}$
The last step of multiplying the $-1$ into the numerator wasn't necessary, but it is common to do so.
We write the simplified form as follows:
$\frac{1-x}{x+1}$ for $x\ne 3$

### Main takeaway

The factors $x-3$ and $3-x$ are opposites since $-1\cdot \left(x-3\right)=3-x$.
In this example, we saw that these factors canceled, but that a factor of $-1$ was added. In other words, the factors $x-3$ and $3-x$ canceled to $\mathit{\text{-1}}$.
In general opposite factors $a-b$ and $b-a$ will cancel to $-1$ provided that $a\ne b$.

2) Simplify $\frac{\left(x-2\right)\left(x-5\right)}{\left(2-x\right)\left(x+5\right)}$.

3) Simplify $\frac{15-10x}{8{x}^{3}-12{x}^{2}}$.
for $x\ne$

## Let's try some more problems

4) Simplify $\frac{3x}{15{x}^{2}-6x}$.
5) Simplify $\frac{3{x}^{3}-15{x}^{2}+12x}{3x-3}$.
for $x\ne$
6) Simplify $\frac{6{x}^{2}-12x}{6x-3{x}^{2}}$.