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 xx, like 3x26x13x^2-6x-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:
1x\dfrac{1}{x}, x+5x24x+4\quad\dfrac{x+5}{x^2-4x+4}, x(x+1)(2x3)x6\quad\dfrac{x(x+1)(2x-3)}{x-6}
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 2x+3x2\dfrac{2x+3}{x-2}.
We can determine the value of this expression for particular xx-values. For example, let's evaluate the expression at x=1\blueD{x}=\blueD1.
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 x=1\blueD{x}=\blueD{1} is 5\goldD{-5}.
Now let's find the value of the expression at x=2\blueD{x}=\blueD{2}.
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 22 makes the denominator 00. Since division by 00 is undefined, x=2\blueD x=\blueD 2 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 00 (since division by 00 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 x+1(x3)(x+4)\dfrac{x+1}{(x-3)(x+4)}

Let's find the zeros of the denominator and then restrict these values:
(x3)(x+4)=0x3=0orx+4=0Zero product propertyx=3orx=4Solve for x\begin{aligned} &(x-3)(x+4)= 0 \\\\ &x-3=0 \quad \text{or} \quad x+4=0 &&\small{\gray{\text{Zero product property}}}\\\\ &x = 3 \quad\text{or} \quad x=-4 &&\small{\gray{\text{Solve for $x$}}}\end{aligned}
So we write that the domain is all real numbers except 3\textit 3 and -4\textit{-4}, or simply x3,4x\neq 3, -4.

Check your understanding

1) What is the domain of x+1x7\dfrac{x+1}{x-7}?
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:
x7=0x=7Solve for x\begin{aligned} &x-7= 0 \\\\ &x = 7 &&\small{\gray{\text{Solve for $x$}}}\end{aligned}
Since 77 makes the denominator 00, we must exclude this from the domain.
The domain is all real numbers except 77, or simply x7x\neq7
2) What is the domain of 3x72x+1\dfrac{3x-7}{2x+1}?
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:
2x+1=02x=1Subtract  from both sides1x=12Divide both sides by 2\begin{aligned} 2x+1&=0\\\\ 2x& =-1&&\small{\gray{\text{Subtract $1$ from both sides}}}\\\\ x&=- \dfrac{1}{2} &&\small{\gray{\text{Divide both sides by $2$}}}\end{aligned}
Since 12-\dfrac12 makes the denominator 00, we must exclude this from the domain.
The domain is all real numbers except 12-\dfrac12, or simply x12x\neq-\dfrac12.
3) What is the domain of 2x3x(x+1)\dfrac{2x-3}{x(x+1)}?
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:
x(x+1)=0x=0orx+1=0Zero product propertyx=0orx=1Solve for x\begin{aligned} &x(x+1)= 0 \\\\ &x=0 \quad \text{or} \quad x+1=0 &&\small{\gray{\text{Zero product property}}}\\\\ &x = 0 \quad\text{or} \quad x=-1 &&\small{\gray{\text{Solve for $x$}}}\end{aligned}
Since the zeros of the denominator are 00 and 1-1, the domain will exclude these values.
The domain is all real numbers except 00 and 1-1. In other words, x0,1x\neq0, -1.

Challenge problems

4*) What is the domain of x3x22x8\dfrac{x-3}{x^2-2x-8}?
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 x22x8=0x^2-2x-8=0 can be easily factored. So we have:
x22x8=0(x4)(x+2)=0x4=0orx+2=0Zero product propertyx=4orx=2Solve for x\begin{aligned} &x^2-2x-8= 0 \\\\ &(x-4)(x+2)=0\\\\ &x-4=0 \quad \text{or} \quad x+2=0 &&\small{\gray{\text{Zero product property}}}\\\\ &x = 4 \quad\text{or} \quad x=-2 &&\small{\gray{\text{Solve for $x$}}}\end{aligned}
Since the zeros of the denominator are 2-2 and 44, the domain will exclude these values.
The domain is all real numbers except 2-2 and 44. In other words, x2,4x\neq-2, 4.
5*) What is the domain of x+2x2+4\dfrac{x+2}{x^2+4}?
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 x2+4>0x^2+4>0 for all real numbers xx.
Another way to think about this is to actually try to set the denominator equal to zero and solve for xx.
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 00, there are no restrictions on the domain. So, the domain is the set of all real numbers.
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