Operations with rational expressions | Lesson (article)

What are rational expressions?

Rational expressions look like fractions that have variables in their denominators (and often numerators too). For example, $\frac{x^{2}}{x + 3}$ ‍ is a rational expression. Just as we can add, subtract, multiply, and divide fractions, we can perform the four operations on rational expressions.

Rational expressions also represent the division of one polynomial expression by another. For example, $\frac{x^{2}}{x + 3}$ ‍ represents the division of $x^{2}$ ‍ by $x + 3$ ‍, for which we can find a quotient and a remainder.

In this lesson, we'll learn to:

Simplify rational expressions
Add, subtract, multiply, and divide rational expressions
Rewrite rational expressions in the form of quotients and remainders

This lesson builds upon the following skills:

Factoring quadratic and polynomial expressions
Operations with polynomials

You can learn anything. Let's do this!

How do I simplify rational expressions?

Intro to rational expression simplification

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Reducing rational expressions to lowest terms

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Just like fractions, but with polynomial factoring

You're probably familiar with simplifying fractions like $\frac{5}{10}$ ‍; we can factor out a $5$ ‍ from both the numerator and the denominator and cancel them, leaving us with $\frac{1}{2}$ ‍.

$\begin{aligned} \frac{5}{10} & = \frac{1 \cdot 5}{2 \cdot 5} \\ = \frac{1}{2} \end{aligned}$ ‍

With rational expressions, we can also cancel out factors that appear in both the numerator and the denominator. These factors can be polynomials!

For example, we can simplify the rational expression $\frac{x + 1}{2 x + 2}$ ‍ by factoring out $x + 1$ ‍ from both the numerator and the denominator and canceling them, leaving us with $\frac{1}{2}$ ‍.

$\begin{aligned} \frac{x + 1}{2 x + 2} & = \frac{1 \cdot (x + 1)}{2 \cdot (x + 1)} \\ = \frac{1}{2} \end{aligned}$ ‍

On the SAT, the numerators and denominators of rational expressions can also be quadratic expressions and higher order polynomials, so the ability to factor these expressions fluently is key to your success.

Try it!

try: find the simplified rational expression

$\frac{x^{2} - 5 x + 6}{x^{2} + x - 6} = \frac{(x - 2) (x - 3)}{(x - 2) (x + 3)}$ ‍

The rational expression $\frac{x^{2} - 5 x + 6}{x^{2} + x - 6}$ ‍ is factored as shown above. Since both the numerator and the denominator contain

as a factor, we can cancel the identical factors to simplify the expression.

$\frac{x^{2} - 5 x + 6}{x^{2} + x - 6}$ ‍ is equivalent to

How do I multiply and divide rational expressions?

Multiplying & dividing rational expressions: monomials

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Multiplying & dividing rational expressions: monomials

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Multiplying and dividing rational expressions

The same rules for multiplying and dividing fractions apply to multiplying and dividing rational expressions.

When multiplying two rational expressions:

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$ ‍

When dividing two expressions, recall that dividing by a fraction is equivalent to multiplying by that fraction's reciprocal:

$\frac{(\frac{a}{b})}{(\frac{c}{d})} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a d}{b c}$ ‍

However, to avoid lengthy polynomial operations, it's recommended that you factor and cancel any cancellable factors before you write out the final expression.

To multiply two rational expressions:

Factor any factorable polynomial expressions in the numerators and the denominators.
Cancel any identical factors that appear in both the numerators and the denominators of the expressions.
Multiply the remaining numerators and multiply the remaining denominators.

Dividing two rational expressions is similar to multiplying; just remember that dividing by an expression is equivalent to multiplying by the reciprocal of the same expression.

Let's look at some examples!

What is the product of $\frac{x^{2}}{x + 3}$ ‍ and $\frac{x + 3}{x}$ ‍ ?

$\begin{aligned} \frac{x^{2}}{x + 3} \cdot \frac{x + 3}{x} \\ = \frac{x \cdot x}{x + 3} \cdot \frac{x + 3}{x} \\ = x \end{aligned}$ ‍

$x$ ‍ is the product of $\frac{x^{2}}{x + 3}$ ‍ and $\frac{x + 3}{x}$ ‍.

If $f (x) = \frac{x^{2} + 2 x + 1}{x + 3}$ ‍ and $g (x) = \frac{x}{x^{2} + 4 x + 3}$ ‍, what is $\frac{f (x)}{g (x)}$ ‍ ?

Both $x^{2} + 2 x + 1$ ‍ and $x^{2} + 4 x + 3$ ‍ can be factored:

$x^{2} + 2 x + 1 = (x + 1) (x + 1)$ ‍
$x^{2} + 4 x + 3 = (x + 1) (x + 3)$ ‍

Dividing by a rational expression is equivalent to multiplying by its reciprocal:

$\begin{aligned} \frac{f (x)}{g (x)} & = \frac{(\frac{x^{2} + 2 x + 1}{x + 3})}{(\frac{x}{x^{2} + 4 x + 3})} \\ = \frac{x^{2} + 2 x + 1}{x + 3} \cdot \frac{x^{2} + 4 x + 3}{x} \\ = \frac{(x + 1) (x + 1)}{x + 3} \cdot \frac{(x + 1) (x + 3)}{x} \\ = \frac{(x + 1) (x + 1)}{x + 3} \cdot \frac{(x + 1) (x + 3)}{x} \\ = \frac{(x + 1) (x + 1) (x + 1)}{x} \\ = \frac{(x + 1)^{3}}{x} \end{aligned}$ ‍

$\frac{f (x)}{g (x)}$ ‍ is $\frac{(x + 1)^{3}}{x}$ ‍.

Try it!

try: identify factorable numerators and denominators

$\frac{x^{2} + 1}{x + 1} \cdot \frac{x^{2} + 3 x + 2}{3 x^{2} + 3}$ ‍

Before multiplying the two rational expressions shown above, Raj wants to factor the numerators and denominators. Which of the following can be factored?

Choose all answers that apply:

(Choice A)
$x^{2} + 1$ ‍
(Choice B)
$x + 1$ ‍
(Choice C)
$x^{2} + 3 x + 2$ ‍
(Choice D)
$3 x^{2} + 3$ ‍

How do I add and subtract rational expressions?

Adding rational expressions with different denominators

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Adding rational expression: unlike denominators

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Adding and subtracting rational expressions

The same rules for adding and subtracting fractions apply to adding and subtracting rational expressions.

When adding or subtracting two rational expressions with unlike denominators:

$\frac{a}{b} \pm \frac{c}{d} = \frac{a d}{b d} \pm \frac{b c}{b d} = \frac{a d \pm b c}{b d}$ ‍

Remember that you can only add and subtract the numerators of the rational expressions if the expressions have a common denominator! In most cases, the easiest way to find a common denominator is to multiply the two unlike denominators, $b$ ‍ and $d$ ‍.

To add and subtract two rational expressions:

Find a common denominator for the two expressions. In most cases, the product of the two denominators would work.
Rewrite the equivalent form of each rational expression using the common denominator.
Add or subtract the numerators of the expressions while retaining the common denominator.
Combine like terms and write the result.
Factor and/or cancel as needed.

Let's look at some examples!

What is the sum of $\frac{x^{2}}{x + 3}$ ‍ and $\frac{4 x + 3}{x + 3}$ ‍ ?

Since the two rational expressions have the same denominator, we can add the numerators and keep the denominator the same.

$\begin{aligned} \frac{x^{2}}{x + 3} + \frac{4 x + 3}{x + 3} \\ = \frac{x^{2} + 4 x + 3}{x + 3} \\ = \frac{(x + 1) (x + 3)}{x + 3} \\ = \frac{(x + 1) (x + 3)}{x + 3} \\ = x + 1 \end{aligned}$ ‍

$x + 1$ ‍ is the sum of $\frac{x^{2}}{x + 3}$ ‍ and $\frac{4 x + 3}{x + 3}$ ‍.

What is the difference $\frac{2 x}{x + 3} - \frac{3}{x + 1}$ ‍ ?

We need to find a common denominator for $\frac{2 x}{x + 3}$ ‍ and $- \frac{3}{x + 1}$ ‍. One common denominator is simply the product of the denominators, $(x + 3) (x + 1)$ ‍.

To rewrite both rational expressions with the common denominator, we multiply the numerator and denominator of each rational expression by the denominator of the other expression:

$\begin{aligned} \frac{2 x}{x + 3} - \frac{3}{x + 1} \\ = \frac{2 x}{x + 3} \cdot \frac{x + 1}{x + 1} - \frac{3}{x + 1} \cdot \frac{x + 3}{x + 3} \\ = \frac{2 x (x + 1) - 3 (x + 3)}{(x + 3) (x + 1)} \\ = \frac{(2 x) (x) + (2 x) (1) + (- 3) (x) + (- 3) (3)}{(x) (x) + (x) (1) + (3) (x) + (3) (1)} \\ = \frac{2 x^{2} + 2 x - 3 x - 9}{x^{2} + x + 3 x + 3} \\ = \frac{2 x^{2} - x - 9}{x^{2} + 4 x + 3} \end{aligned}$ ‍

$\frac{2 x^{2} - x - 9}{x^{2} + 4 x + 3}$ ‍ is the difference $\frac{2 x}{x + 3} - \frac{3}{x + 1}$ ‍.

Try it!

try: rewrite expressions with common denominators

Celeste wants to find the sum of $\frac{3}{x}$ ‍ and $\frac{x}{2 x - 3}$ ‍. In order to add the two rational expressions, she must first find a common denominator by

the two denominators, $x$ ‍ and $2 x - 3$ ‍.

Next, she needs to rewrite each rational expression using the common denominator. The first term can be rewritten as $\frac{3 (2 x - 3}{x (2 x - 3)}$ ‍, and the second term can be rewritten as

How do I rewrite a rational expression as a quotient and a remainder?

Dividing polynomials by linear expressions

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Dividing polynomials by linear expressions

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Polynomial long division

We can represent any rational expression as $\frac{a (x)}{b (x)}$ ‍, where $a$ ‍ and $b$ ‍ are polynomial expressions in terms of $x$ ‍.

For example, for $a (x) = x^{2} + 2 x + 4$ ‍ and $b (x) = x + 3$ ‍, $\frac{a (x)}{b (x)} = \frac{x^{2} + 2 x + 4}{x + 3}$ ‍.

When dividing $a$ ‍ and $b$ ‍, we can find quotient polynomial $q$ ‍ and remainder polynomial $r$ ‍ such that:

$\frac{a (x)}{b (x)} = q (x) + \frac{r (x)}{b (x)}$ ‍

Where the degree of $r$ ‍ is less than the degree of $b$ ‍. Since $b$ ‍ is usually a first degree polynomial ( $a x + b$ ‍) on the SAT, $r$ ‍ is usually a constant.

When dividing two polynomials using long division, we focus on the highest degree terms in the numerator and the denominator first. For example, for $\frac{x^{2} + 2 x + 4}{x + 3}$ ‍, the highest degree term in the numerator is $x^{2}$ ‍, and the highest degree term in the denominator is $x$ ‍. The first question we ask is "what is $x^{2}$ ‍ divided by $x$ ‍ ?"

$\frac{x^{2}}{x} = x$ ‍, so we write $x$ ‍ as the first term of the quotient $q$ ‍, find the product of $x$ ‍ and the divisor $x + 3$ ‍, then subtract the product from $a$ ‍. This eliminates the $x^{2}$ ‍ term in the dividend.

$\begin{aligned} x \\ x + 3 | & \overset{―}{x^{2} + 2 x + 4} \\ - ( & x^{2} + 3 x) \\ \overset{―}{- x + 4} \end{aligned}$ ‍

Next, we do the same to what's left of the dividend, $- x + 4$ ‍. We ask "what is $- x$ ‍ divided by $x$ ‍ ?"

$\frac{- x}{x} = - 1$ ‍, so we write $- 1$ ‍ as the second term of the quotient $q$ ‍, find the product of $- 1$ ‍ and the divisor $x + 3$ ‍, then subtract the product from $- x + 4$ ‍. This eliminates the $x$ ‍ term in the dividend.

$\begin{aligned} x - 1 \\ x + 3 | & \overset{―}{x^{2} + 2 x + 4} \\ - ( & x^{2} + 3 x) \\ \overset{―}{- x + 4} \\ - & (- x - 3) \\ \overset{―}{7} \end{aligned}$ ‍

This leaves us with the constant $7$ ‍. Since the degree of $7$ ‍ is lower than the degree of $x + 3$ ‍, we can stop dividing here and write our quotient and remainder.

$q (x) = x - 1$ ‍
$r (x) = 7$ ‍

Therefore, $\frac{x^{2} + 2 x + 4}{x + 3} = x - 1 + \frac{7}{x + 3}$ ‍.

Another strategy to find the quotient and the remainder is to group the numerator, which requires us to split the numerator of a rational expression into a polynomial divisible by the denominator and the remainder.

You don't need to know how to group the numerator, but it may save you time on the test.

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Dividing quadratics by linear expressions with remainders

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Dividing quadratics by linear expressions with remainders: missing x-term

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To divide polynomial expressions $a (x)$ ‍ and $b (x)$ ‍ using long division:

Divide the highest degree term of $a$ ‍ by the highest degree term of $b$ ‍. This gives you a term of the quotient.
Multiply the result of Step 1 by $b$ ‍.
Subtract the result of Step 2 from $a$ ‍. Be careful when subtracting negatives!
Repeat the divide-multiply-subtract steps using what's left of the dividend until the result is of a lower degree than $b$ ‍.
The terms calculated in the "divide" steps form the quotient $q$ ‍. The leftover polynomial with a lower degree than $b$ ‍ is the remainder $r$ ‍.
Write the result as $q (x) + \frac{r (x)}{b (x)}$ ‍.

Example: For $f (x) = x^{2}$ ‍ and $g (x) = x + 3$ ‍, rewrite $\frac{f (x)}{g (x)}$ ‍ in the form $q (x) + \frac{r (x)}{g (x)}$ ‍.

Let's divide $f (x)$ ‍ by $g (x)$ ‍ to find the quotient and remainder.

First, we can divide $x^{2}$ ‍ by $x$ ‍ to get $x$ ‍.

$\begin{aligned} x \\ x + 3 | & \overset{―}{x^{2} + 0 x} \\ - ( & x^{2} + 3 x) \\ \overset{―}{- 3 x} \end{aligned}$ ‍

Keep in mind that multiplying $x$ ‍ by $x + 3$ ‍ creates an $x$ ‍-term, $3 x$ ‍. Since $x^{2}$ ‍ has no $x$ ‍-term, we're effectively subtracting $3 x$ ‍ from $0 x$ ‍.

Next, we can divide $- 3 x$ ‍ by $x$ ‍ to get $- 3$ ‍:

$\begin{aligned} x - 3 \\ x + 3 | & \overset{―}{x^{2} + 0 x} \\ - ( & x^{2} + 3 x) \\ \overset{―}{- 3 x + 0} \\ - & (- 3 x - 9) \\ \overset{―}{9} \end{aligned}$ ‍

Similarly, multiplying $- 3$ ‍ by $x + 3$ ‍ creates a constant term, $- 9$ ‍. Since $- 3 x$ ‍ has no constant term, we're effectively subtracting $- 9$ ‍ from (or adding $9$ ‍ to) $0$ ‍.

Since our divisor is a first degree term and only a constant term remains, $r$ ‍ is of a lower degree than $g$ ‍, and we can stop dividing. The quotient is $x - 3$ ‍, and the remainder is $9$ ‍. We can write the expression as $x - 3 + \frac{9}{x + 3}$ ‍.

$\frac{f (x)}{g (x)} = x - 3 + \frac{9}{x + 3}$ ‍

Try it!

Try: take the first steps of polynomial long division

$\frac{6 x^{3} + 3 x^{2} - 14 x - 2}{2 x + 1}$ ‍

In the rational expression above, the numerator is a

polynomial and the denominator is a first degree polynomial. Therefore, the quotient must be a

polynomial.

To find the first term of the quotient, we must divide the highest degree term in the numerator,

, by the highest degree term in the denominator, $2 x$ ‍.

The first term of the quotient is

Your turn!

Practice: multiply two rational expressions

Which of the following is equivalent to $\frac{2}{x + 1} \cdot \frac{x + 1}{x + 2}$ ‍ ?

Choose 1 answer:

(Choice A)
$\frac{2}{x + 1}$ ‍
(Choice B)
$\frac{2}{x + 2}$ ‍
(Choice C)
$\frac{x + 3}{2 x + 3}$ ‍
(Choice D)
$\frac{2 x + 2}{2 x + 3}$ ‍

Practice: multiply two rational expressions

$\frac{x^{2} + 5 x + 6}{x + 4} \cdot \frac{x^{2} + 4 x}{x + 3}$ ‍

Which of the following is equivalent to the expression above?

Choose 1 answer:

(Choice A)
$\frac{x}{x + 2}$ ‍
(Choice B)
$\frac{x + 2}{x}$ ‍
(Choice C)
$x + 2$ ‍
(Choice D)
$x^{2} + 2 x$ ‍

practice: subtract two rational expressions

$\frac{4}{x + 1} - \frac{3}{2 x - 3}$ ‍

Which of the following is equivalent to the expression above?

Choose 1 answer:

(Choice A)
$\frac{1}{4 - x}$ ‍
(Choice B)
$\frac{1}{2 x^{2} - x - 3}$ ‍
(Choice C)
$\frac{5 x - 4}{2 x^{2} - x - 3}$ ‍
(Choice D)
$\frac{5 (x - 3)}{2 x^{2} - x - 3}$ ‍

practice: rewrite a rational expression as quotient and remainder

Which of the following is equivalent to $\frac{x^{2} + 5 x + 2}{x + 5}$ ‍ ?

Choose 1 answer:

(Choice A)
$x + 1$ ‍
(Choice B)
$x + 2$ ‍
(Choice C)
$x + \frac{2}{x + 5}$ ‍
(Choice D)
$x + 1 - \frac{4}{x + 5}$ ‍

Operations with rational expressions | Lesson (article) | Khan Academy (2024)

FAQs

What are the operations of a rational expression? ›

If f is a rational expression then f can be written in the form p/q where p and q are polynomials. Like polynomials or any other type of expression, the basic arithmetic operations, namely addition, subtraction, multiplication, and division, can be performed on rational expressions.

Learn More ›

What is the function of a rational expression? ›

A rational expression is an expression of the form pq, where p and q are polynomials and q≠0. A rational function is a function of the form R(x)=p(x)q(x) where p(x) and q(x) are polynomial functions and q(x) is not zero.

View Details ›

How to add rational expressions? ›

To add or subtract two rational expressions with the same denominator, we simply add or subtract the numerators and write the result over the common denominator. When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator.

How to simplify operations with rational expressions? ›

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.

View Details ›

How do you explain rational expressions? ›

A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials.

Show Me More ›

How to identify if it is a rational algebraic expression or not? ›

A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x² + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4.

Find Out More ›

How to tell if a rational expression is proper or improper? ›

Like fractions, rational expressions can be proper or improper. A proper rational expression has a lower-degree numerator than denominator ( e . g . , 1 − x / x ² + 3 ) , and an improper one has a higher-degree numerator than denominator ( e . g . , x ² + 3 / 1 − x ) .

See Details ›

How can rational expressions be used in real life? ›

Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule. A “work problem” is an example of a real life situation that can be modeled and solved using a rational equation.

Learn More Now ›

What is an example of a rational expression? ›

Here are some examples of rational expressions: (x + 1) / (x² - 5), (x³ + 3x² - 5) / (4x - 2), etc. Note that if one of the numerator and denominator is NOT a polynomial, then the fraction is NOT called as a rational expression.

Know More ›

What is the difference between a rational equation and a rational expression? ›

A rational equation has an = symbol, in it we try to solve for a variable, a rational expression does not have an equal sign, with it we simplify perform the indicated operation, addition, subtraction, multiplication or division.

Learn More ›

How to solve operations on rational algebraic expressions? ›

How do I multiply and divide rational expressions?

Factor any factorable polynomial expressions in the numerators and the denominators.
Cancel any identical factors that appear in both the numerators and the denominators of the expressions.
Multiply the remaining numerators and multiply the remaining denominators.

Operations with rational expressions | Lesson (article) | Khan Academy (2024)

What are rational expressions?

How do I simplify rational expressions?

Intro to rational expression simplification

Just like fractions, but with polynomial factoring

Try it!

How do I multiply and divide rational expressions?

Multiplying & dividing rational expressions: monomials

Multiplying and dividing rational expressions

Let's look at some examples!

Try it!

How do I add and subtract rational expressions?

Adding rational expressions with different denominators

Adding and subtracting rational expressions

Let's look at some examples!

Try it!

How do I rewrite a rational expression as a quotient and a remainder?

Dividing polynomials by linear expressions

Polynomial long division

Try it!

Your turn!

FAQs

What are the operations of a rational expression? ›

How to solve operations on rational algebraic expressions? ›