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

Rational expressions look like fractions that have variables in their denominators (and often numerators too). For example, ${\displaystyle \frac{x^2}{x+3}}$‍ 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, ${\displaystyle \frac{x^2}{x+3}}$‍ 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:

1. Simplify rational expressions
2. Add, subtract, multiply, and divide rational expressions
3. 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!

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

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.

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

When multiplying two rational expressions:

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

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:

1. Factor any factorable polynomial expressions in the numerators and the denominators.
2. Cancel any identical factors that appear in both the numerators and the denominators of the expressions.
3. 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.

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

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

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:

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:

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

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

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

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

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

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

Where the degree of $r$‍ is less than the degree of $b$‍. Since $b$‍ is usually a first degree polynomial ($ax+b$‍) 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 ${\displaystyle \frac{x^2+2x+4}{x+3}}$‍, 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 $x$‍ ?"

${\displaystyle \frac{x^2}{x}}=x$‍, 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.

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

${\displaystyle \frac{-x}{x}}=-1$‍, 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.

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

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.

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

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

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