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, start superscript, 2, end superscript, minus, 6, x, minus, 1.
A rational expression is simply a ratio 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, space, start fraction, x, plus, 5, divided by, x, start superscript, 2, end superscript, minus, 4, x, plus, 4, end fraction, space, 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 blueD, x, end color blueD, equals, start color blueD, 1, end color blueD.
2(1)+312=  51=5\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 blueD, x, end color blueD, equals, start color blueD, 1, end color blueD is start color goldD, minus, 5, end color goldD.
Now let's find the value of the expression at start color blueD, x, end color blueD, equals, start color blueD, 2, end color blueD.
2(2)+322=70=undefined!\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 blueD, x, end color blueD, equals, start color blueD, 2, end color blueD 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 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:
So we write that the domain is all real numbers except 3 and negative, 4, or simply x, does not equal, 3, comma, minus, 4.

Check your understanding

1) What is the domain of start fraction, x, plus, 1, divided by, x, minus, 7, end fraction?
Choose 1 answer:
Choose 1 answer:

The domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
Let's find the zeros of the denominator:
Since 7 makes the denominator 0, we must exclude this from the domain.
The domain is all real numbers except 7, or simply x, does not equal, 7
2) What is the domain of start fraction, 3, x, minus, 7, divided by, 2, x, plus, 1, end fraction?
Choose 1 answer:
Choose 1 answer:

The domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
Let's find the zeros of the denominator:
Since minus, start fraction, 1, divided by, 2, end fraction makes the denominator 0, we must exclude this from the domain.
The domain is all real numbers except minus, start fraction, 1, divided by, 2, end fraction, or simply x, does not equal, minus, start fraction, 1, divided by, 2, end fraction.
3) What is the domain of start fraction, 2, x, minus, 3, divided by, x, left parenthesis, x, plus, 1, right parenthesis, end fraction?
Choose 1 answer:
Choose 1 answer:

The domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
Let's find the zeros of the denominator:
Since the zeros of the denominator are 0 and minus, 1, the domain will exclude these values.
The domain is all real numbers except 0 and minus, 1. In other words, x, does not equal, 0, comma, minus, 1.

Challenge problems

4*) What is the domain of start fraction, x, minus, 3, divided by, x, start superscript, 2, end superscript, minus, 2, x, minus, 8, end fraction?
Choose 1 answer:
Choose 1 answer:

The domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
In this case, the denominator is not in factored form. So, we must either factor the denominator or use another method to find its zeros.
Notice that x, start superscript, 2, end superscript, minus, 2, x, minus, 8, equals, 0 can be easily factored. So we have:
Since the zeros of the denominator are minus, 2 and 4, the domain will exclude these values.
The domain is all real numbers except minus, 2 and 4. In other words, x, does not equal, minus, 2, comma, 4.
5*) What is the domain of start fraction, x, plus, 2, divided by, x, start superscript, 2, end superscript, plus, 4, end fraction?
Choose 1 answer:
Choose 1 answer:

The domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
However, in this case the denominator will never be equal to zero. This is because x, start superscript, 2, end superscript, plus, 4, is greater than, 0 for all real numbers x.
Another way to think about this is to actually try to set the denominator equal to zero and solve for x.
x2+4=0x2=4A real number squared is always positive!\begin{aligned} x^2+4&= 0 \\\\ x^2&=-4 &&\small{\gray{\text{A real number squared is always positive!}}}\\\\ \end{aligned}
But here we see that this is impossible because the square of a real number can never be negative!
Because the denominator cannot be 0, there are no restrictions on the domain. So, the domain is the set of all real numbers.