Main content

### Course: Algebra (all content) > Unit 13

Lesson 2: Simplifying rational expressions- Reducing rational expressions to lowest terms
- Reducing rational expressions to lowest terms
- Reduce rational expressions to lowest terms: Error analysis
- Simplifying rational expressions: common binomial factors
- Simplifying rational expressions: opposite common binomial factors
- Simplifying rational expressions (advanced)
- Reduce rational expressions to lowest terms
- Simplifying rational expressions: grouping
- Simplify rational expressions (advanced)
- Simplifying rational expressions (old video)

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Simplifying rational expressions (advanced)

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 $\text{}{\displaystyle \frac{10{x}^{3}}{2{x}^{2}-18x}}$

**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.

**Step 2: List restricted values**

From the factored form, we see thatand $x\ne 0$ . $x\ne 9$

**Step 3: Cancel common factors**

**Step 4: Final answer**

We write the simplified form as follows:

for $\frac{5{x}^{2}}{x-9}$ $x\ne 0$

### Main takeaway

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

### Check your understanding

## Example 2: Simplifying $\text{}{\displaystyle \frac{(3-x)(x-1)}{(x-3)(x+1)}}$

**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 $x-3$ .

**Step 2: List restricted values**

From the factored form, we see that $x\ne 3$ and $x\ne -1$ .

**Step 3: Cancel common factors**

The last step of multiplying the $-1$ into the numerator wasn't necessary, but it is common to do so.

**Step 4: Final answer**

We write the simplified form as follows:

### Main takeaway

The factors $x-3$ and $3-x$ are $-1\cdot (x-3)=3-x$ .

**opposites**sinceIn this example, we saw that these factors canceled, but that a factor of $-1$ was added. In other words, the factors $x-3$ and $3-x$ $\mathit{\text{-1}}$ .

*canceled to*In general opposite factors $a-b$ and $b-a$ will cancel to $-1$ provided that $a\ne b$ .

### Check your understanding

## Let's try some more problems

## Want to join the conversation?

No posts yet.