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Reducing rational expressions to lowest terms

Learn what it means to reduce a rational expression to lowest terms, and how it's done!

What you should be familiar with before taking this lesson

A rational expression is a ratio of two polynomials. The domain of a rational expression is all real numbers except those that make the denominator equal to zero.
For example, the domain of the rational expression $\frac{x+2}{x+1}$ is all real numbers except $\mathit{\text{-1}}$, or $x\ne -1$.
If this is new to you, we recommend that you check out our intro to rational expressions.
You should also know how to factor polynomials for this lesson.

What you will learn in this lesson

In this article, we will learn how to reduce rational expressions to lowest terms by looking at several examples.

Introduction

A rational expression is reduced to lowest terms if the numerator and denominator have no factors in common.
We can reduce rational expressions to lowest terms in much the same way as we reduce numerical fractions to lowest terms.
For example, $\frac{6}{8}$ reduced to lowest terms is $\frac{3}{4}$. Notice how we canceled a common factor of $2$ from the numerator and the denominator:
$\begin{array}{rl}\frac{6}{8}& =\frac{2\cdot 3}{2\cdot 4}\\ \\ & =\frac{\overline{)2}\cdot 3}{\overline{)2}\cdot 4}\\ \\ & =\frac{3}{4}\end{array}$

Example 1: Reducing $\frac{{x}^{2}+3x}{{x}^{2}+5x}$‍  to lowest terms

Step 1: Factor the numerator and denominator
The only way to see if the numerator and denominator share common factors is to factor them!
$\frac{{x}^{2}+3x}{{x}^{2}+5x}=\frac{x\left(x+3\right)}{x\left(x+5\right)}$
Step 2: List restricted values
At this point, it is helpful to notice any restrictions on $x$. These will carry over to the simplified expression.
Since division by $0$ is undefined, here we see that $x\ne 0$ and $x\ne -5$.
$\frac{x\left(x+3\right)}{x\left(x+5\right)}$
Step 3: Cancel common factors
Now notice that the numerator and denominator share a common factor of $x$. This can be canceled out.
$\begin{array}{rl}\frac{x\left(x+3\right)}{x\left(x+5\right)}& =\frac{\overline{)x}\left(x+3\right)}{\overline{)x}\left(x+5\right)}\\ \\ & =\frac{x+3}{x+5}\end{array}$
Recall that the original expression is defined for $x\ne 0,-5$. The reduced expression must have the same restrictions.
Because of this, we must note that $x\ne 0$. We do not need to note that $x\ne -5$, since this is understood from the expression.
In conclusion, the reduced form is written as follows:
$\frac{x+3}{x+5}$ for $x\ne 0$

Original expressionReduced expression
$\frac{{x}^{2}+3x}{{x}^{2}+5x}$$\frac{x+3}{x+5}$ for $x\ne 0$
The two expressions above are equivalent. This means their outputs are the same for all possible $x$-values. The table below illustrates this for $x=2$.
Original expressionReduced expression
Evaluation at $x=2$$\begin{array}{rl}\frac{\left(2{\right)}^{2}+3\left(2\right)}{\left(2{\right)}^{2}+5\left(2\right)}& =\frac{10}{14}\\ \\ & =\frac{2\cdot 5}{2\cdot 7}\\ \\ & =\frac{\overline{)2}\cdot 5}{\overline{)2}\cdot 7}\\ \\ & =\frac{5}{7}\end{array}$$\begin{array}{rl}\frac{2+3}{2+5}& =\frac{5}{7}\\ \\ & \phantom{=\frac{5}{7}}\\ \\ & \phantom{=\frac{5}{7}}\\ \\ & \phantom{=\frac{5}{7}}\end{array}$
NoteThe result is reduced to lowest terms by canceling a common factor of $2$.The result is already reduced to lowest terms because the factor of $x$ (in this case $x=2$), was already canceled when we reduced the expression to lowest terms.
For this reason, the two expressions have the same value for the same input. However, values that make the original expression undefined often break this rule. Notice how this is the case with $x=0$.
Original expressionReduced expression (without restriction)
Evaluation at $x=0$$\begin{array}{rl}\frac{\left(0{\right)}^{2}+3\left(0\right)}{\left(0{\right)}^{2}+5\left(0\right)}& =\frac{0}{0}\\ \\ & =\text{undefined}\end{array}$$\begin{array}{rl}\frac{0+3}{0+5}& =\frac{3}{5}\\ \\ & \phantom{\text{undefined}}\end{array}$
Because the two expressions must be equivalent for all possible inputs, we must require $x\ne 0$ for the reduced expression.

Note that we cannot cancel the $x$'s in the expression below. This is because these are terms rather than factors of the polynomials!
$\frac{x+3}{x+5}\ne \frac{3}{5}$
This becomes clear when looking at a numerical example. For example, suppose $x=2$.
$\frac{2+3}{2+5}\ne \frac{3}{5}$
As a rule, we can only cancel if the numerator and denominator are in factored form!

Summary of the process for reducing to lowest terms

• Step 1: Factor the numerator and the denominator.
• Step 2: List restricted values.
• Step 3: Cancel common factors.
• Step 4: Reduce to lowest terms and note any restricted values not implied by the expression.

Problem 1
Reduce $\frac{6x+20}{2x+10}$ to lowest terms.

Problem 2
Reduce $\frac{{x}^{3}-3{x}^{2}}{4{x}^{2}-5x}$ to lowest terms.
for $x\ne$

Example 2: Reducing $\frac{{x}^{2}-9}{{x}^{2}+5x+6}$‍  to lowest terms

Step 1: Factor the numerator and denominator
$\frac{{x}^{2}-9}{{x}^{2}+5x+6}=\frac{\left(x-3\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}$
Step 2: List restricted values
Since division by $0$ is undefined, here we see that $x\ne -2$ and $x\ne -3$.
$\frac{\left(x-3\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}$
Step 3: Cancel common factors
Notice that the numerator and denominator share a common factor of $x+3$. This can be canceled out.
$\begin{array}{rl}\frac{\left(x-3\right)\left(x+3\right)}{\left(x+2\right)\left(x+3\right)}& =\frac{\left(x-3\right)\overline{)\left(x+3\right)}}{\left(x+2\right)\overline{)\left(x+3\right)}}\\ \\ & =\frac{x-3}{x+2}\end{array}$
We write the reduced form as follows:
$\frac{x-3}{x+2}$ for $x\ne -3$
The original expression requires $x\ne -2,-3$. We do not need to note that $x\ne -2$, since this is understood from the expression.

Check for understanding

Problem 3
Reduce $\frac{{x}^{2}-3x+2}{{x}^{2}-1}$ to lowest terms.

Problem 4
Reduce $\frac{{x}^{2}-2x-15}{{x}^{2}+x-6}$ to lowest terms.
for $x\ne$

Want to join the conversation?

• Will I need this in real life?
• Maybe.
Rational functions appear quite often in business and economics applications.

Additionally, understanding rational functions will help you to understand more complex functions that can model other "real life" situations.
Most people never notice when they can model a situation using rational function (or other mathematical functions) because they never attempt to do so.

Hopefully you will keep sincerely asking questions like this and looking for ways to use this knowledge to model things in the real world. Such an outlook will serve you well.
• This is very confusing? How will this help us in real life?
• It probably, most likely, won't. although some parts of the business and trade use this math, although I don't know any, there might be some.
• I don't understand much the restricted values. Can someone help me to understand it please?
• If we have an equation (x^2 + 3x) / (x^2 + 5x) , we can simplify it to (x+3) / (x+5) . At first glance, the two equations seem to be equal, but they are actually not!
Remember that x can be any value - let just try the values -1, 1 and 0 for x in the two equations above.
For x = -1,
(x^2 + 3x) / (x^2 + 5x) = (1 - 3) / (1 - 5) = -2/-4 = 0.5
(x+3) / (x+5) = (-1 + 3)/ (-1 + 5) = 2/4 = 0.5
Both equations are equivalent as both of them give the value 0.5
For x = 1,
(x^2 + 3x) / (x^2 + 5x) = (1 + 3) / (1 + 5) = 4/6 = 2/3
(x+3) / (x+5) = (1 + 3)/ (1 + 5) = 4/6 = 2/3
Both equations are equivalent as both of them give the value 2/3
For x = 0,
(x^2 + 3x) / (x^2 + 5x) = (0 + 0) / (0 + 0) = undefined !
(x+3) / (x+5) = (0 + 3)/ (0 + 5) = 3/5 !
Both equations are NOT equivalent as both of them give different values in the case of x =0
So from the above, whenever the denominator results in a value of 0, we get an undefined value for the expression. Therefore we need to put the restriction x not equal to 0 in this case on the equation (x+3) / (x+5). With this restriction, the simplified equation is now equivalent to the original rational expression.
• How do I remember all of these steps?
• Practice, practice, practice. Eventually, you will do it enough that you don’t even think about it. You could come up with an acronym to help you, but the final goal is that you won’t need it.
• who invented this torture
• onb bruh
• In #3, why can't x be equal to 1? I understand why it can't equal -1, but I don't see where 1 causes the expression to be undefined.
• Factor both the numerator and the denominator.
Numerator = (x-2)(x-1)
Denominator = (x-1)(x+1)
The factor of (x-1) in the denominator would cause the an undefined if x=1
Even though this factor cancels out, you need to maintain the restriction that x can't = 1 to be consistent with the original expression.
Hope this helps.
• what makes an expression undefined?
• With rational expressions (fractions), they become undefined if the denominator = 0. If a denominator = 0, then we are dividing by 0, which is undefined.
• Why is x not equal to -3? And why is that stated if we found that out in the middle of the problem?
• Your simplified fraction must be equivalent to the original fraction. The original fraction is undefined for x=-3 and x=2. Once the fraction is reduced, you can no longer tell that the new fraction needs to be undefined for x=-3, so it must be explicitly stated. If you don't state it, you omit one of the restrictions that existed in the original fraction so the new fraction is no longer equivalent to the original.

Hope this helps.
• In Example 1 if x= -5 at any step this would make the whole expression undefined giving a zero in the denominator. Why is only x can not equal 0 listed and not x can not equal -5?
• You are correct. But Sal is concerned about the now missing (after x/x is eliminated) x in the denominator, and not (x+5), which is still there to see.
• Just to get this clear, the way you figure out restricted values is like this, right?:
Let's say you have and equation that you simplified to q+1/q+2.
The restriction is that q cannot be equal to -2, correct?
I'm studying for SSATs, so I need to make sure I know this stuff. It's always been a little confusing, but now I think I've got it. Just need this clarified. Also, when in the article it said that "We do not need to note that x≠−5, since this is understood from the expression." Why don't we need to note that? How is it understood? Why wouldn't you list all the restrictions, instead of just one?