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## Precalculus

### Course: Precalculus>Unit 4

Lesson 1: Reducing rational expressions to lowest terms

# Intro to rational expressions

Learn what rational expressions are and about the values for which they are undefined.

#### What you will learn in this lesson

This lesson will introduce you to rational expressions. You will learn how to determine when a rational expression is undefined and how to find its domain.

## What is a rational expression?

A polynomial is an expression that consists of a sum of terms containing integer powers of x, like 3, x, squared, minus, 6, x, minus, 1.
A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials.
These are examples of rational expressions:
• start fraction, 1, divided by, x, end fraction
• start fraction, x, plus, 5, divided by, x, squared, minus, 4, x, plus, 4, end fraction
• start fraction, x, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, 2, x, minus, 3, right parenthesis, divided by, x, minus, 6, end fraction
Notice that the numerator can be a constant and that the polynomials can be of varying degrees and in multiple forms.

## Rational expressions and undefined values

Consider the rational expression start fraction, 2, x, plus, 3, divided by, x, minus, 2, end fraction.
We can determine the value of this expression for particular x-values. For example, let's evaluate the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 1, end color #11accd.
\begin{aligned} \dfrac{2(\blueD{1})+3}{\blueD1-2} &= \dfrac{5}{-1} \\\\ &=\goldD{-5} \end{aligned}
From this, we see that the value of the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 1, end color #11accd is start color #e07d10, minus, 5, end color #e07d10.
Now let's find the value of the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 2, end color #11accd.
\begin{aligned} \dfrac{2(\blueD{2})+3}{\blueD2-2} &= \dfrac{7}{0} \\\\ &=\goldD{\text{undefined!}} \end{aligned}
An input of 2 makes the denominator 0. Since division by 0 is undefined, start color #11accd, x, end color #11accd, equals, start color #11accd, 2, end color #11accd is not a possible input for this expression!

## Domain of rational expressions

The domain of any expression is the set of all possible input values.
In the case of rational expressions, we can input any value except for those that make the denominator equal to 0 (since division by 0 is undefined).
In other words, the domain of a rational expression includes all real numbers except for those that make its denominator zero.

### Example: Finding the domain of $\dfrac{x+1}{(x-3)(x+4)}$start fraction, x, plus, 1, divided by, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 4, right parenthesis, end fraction

Let's find the zeros of the denominator and then restrict these values:
\begin{aligned} (x-3)&(x+4)= 0 \\\\ \swarrow \quad&\quad \searrow \\\\ x-3=0 \quad &\text{or} \quad x+4=0 \\\\ x = 3 \quad&\text{or} \quad x=-4 \end{aligned}
So we write that the domain is all real numbers except start text, 3, end text and start text, negative, 4, end text, or simply x, does not equal, 3, comma, minus, 4.

Problem 1
What is the domain of start fraction, x, plus, 1, divided by, x, minus, 7, end fraction?

Problem 2
What is the domain of start fraction, 3, x, minus, 7, divided by, 2, x, plus, 1, end fraction?

Problem 3
What is the domain of start fraction, 2, x, minus, 3, divided by, x, left parenthesis, x, plus, 1, right parenthesis, end fraction?