Lesson 1.6: Rational Expressions

New Concept

A rational expression is a fraction with polynomials in the numerator and denominator.
This lesson covers how to perform arithmetic operations—simplifying, multiplying, dividing, adding, and subtracting—on these expressions by factoring and finding common denominators.

What’s next

This card sets the foundation. Soon, you'll master each operation through a series of interactive examples, practice cards, and challenge problems on our platform.

Property

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator.

To simplify a rational expression:

1. Factor the numerator and denominator.
2. Cancel any common factors.

Examples

• Simplify $\frac{x^2 - 4}{x^2 + 3x + 2}$. We factor it to $\frac{(x - 2)(x + 2)}{(x + 1)(x + 2)}$. Canceling the common factor $(x + 2)$ gives us $\frac{x - 2}{x + 1}$.
• Simplify $\frac{y^2 + 6y + 9}{y^2 - 9}$. Factoring gives $\frac{(y + 3)^2}{(y + 3)(y - 3)}$. After canceling $(y + 3)$, the simplified form is $\frac{y + 3}{y - 3}$.
• Simplify $\frac{3a^2 - 3}{6a + 6}$. First, factor out numbers and expressions: $\frac{3(a^2 - 1)}{6(a + 1)} = \frac{3(a - 1)(a + 1)}{6(a + 1)}$. Cancel the 3 and the 6 to get $\frac{1}{2}$, and cancel $(a+1)$ to get $\frac{a - 1}{2}$.

Property

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions.

To multiply two rational expressions:

1. Factor the numerator and denominator.
2. Multiply the numerators.
3. Multiply the denominators.
4. Simplify by canceling common factors.

Examples

• Multiply $\frac{x^2 - 25}{3x + 9} \cdot \frac{x + 3}{x - 5}$. Factoring gives $\frac{(x - 5)(x + 5)}{3(x + 3)} \cdot \frac{x + 3}{x - 5}$. After multiplying and canceling, the result is $\frac{x + 5}{3}$.
• Multiply $\frac{a^2 + 4a + 3}{a^2 - 4} \cdot \frac{a + 2}{a + 1}$. Factor to get $\frac{(a + 3)(a + 1)}{(a - 2)(a + 2)} \cdot \frac{a + 2}{a + 1}$. Cancel common factors to find the product is $\frac{a + 3}{a - 2}$.
• Multiply $\frac{y^2 - y - 20}{y + 2} \cdot \frac{y + 2}{y - 5}$. Factor the numerator to $\frac{(y - 5)(y + 4)}{y + 2} \cdot \frac{y + 2}{y - 5}$. The simplified product is $y + 4$.

Property

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second.

To divide two rational expressions:

1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
2. Factor the numerators and denominators.
3. Multiply the numerators.
4. Multiply the denominators.
5. Simplify.

Examples

• Divide $\frac{x^2 - 16}{x + 1} \div \frac{x + 4}{x + 1}$. Rewrite as $\frac{(x - 4)(x + 4)}{x + 1} \cdot \frac{x + 1}{x + 4}$. Canceling common factors leaves $x - 4$.
• Divide $\frac{a^2 + 4a + 4}{a^2 - 36} \div \frac{a + 2}{a - 6}$. This becomes $\frac{(a + 2)^2}{(a - 6)(a + 6)} \cdot \frac{a - 6}{a + 2}$. The simplified quotient is $\frac{a + 2}{a + 6}$.
• Divide $\frac{3y^2 + 5y - 2}{y^2 - 4} \div \frac{3y - 1}{y + 2}$. Rewrite as $\frac{(3y - 1)(y + 2)}{(y - 2)(y + 2)} \cdot \frac{y + 2}{3y - 1}$. The result is $\frac{y + 2}{y - 2}$.

Property

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions.
We need to find a common denominator.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common.
To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors.

To add or subtract rational expressions:

1. Factor the numerator and denominator.
2. Find the LCD of the expressions.
3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.
4. Add or subtract the numerators.
5. Simplify.

Examples

• Add $\frac{4}{a} + \frac{9}{b}$. The LCD is $ab$. We get $\frac{4b}{ab} + \frac{9a}{ab} = \frac{4b + 9a}{ab}$.
• Subtract $\frac{7}{x^2 - 4} - \frac{3}{x + 2}$. Since $x^2 - 4 = (x-2)(x+2)$, the LCD is $(x-2)(x+2)$. This gives $\frac{7}{(x-2)(x+2)} - \frac{3(x-2)}{(x-2)(x+2)} = \frac{7 - 3x + 6}{(x-2)(x+2)} = \frac{13 - 3x}{x^2 - 4}$.
• Add $\frac{b}{b + 3} + \frac{5}{b - 2}$. The LCD is $(b+3)(b-2)$. The sum is $\frac{b(b-2)}{(b+3)(b-2)} + \frac{5(b+3)}{(b+3)(b-2)} = \frac{b^2 - 2b + 5b + 15}{(b+3)(b-2)} = \frac{b^2 + 3b + 15}{b^2 + b - 6}$.

Property

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing.

To simplify a complex rational expression:

1. Combine expressions in the numerator into a single rational expression.
2. Combine expressions in the denominator into a single rational expression.
3. Rewrite as the numerator divided by the denominator.
4. Rewrite as multiplication by the reciprocal of the denominator.
5. Multiply and simplify.

Examples

• Simplify $\frac{\frac{2}{x} + \frac{3}{y}}{\frac{1}{xy}}$. Combine the numerator to get $\frac{2y + 3x}{xy}$. Now we have $\frac{\frac{2y + 3x}{xy}}{\frac{1}{xy}}$. This simplifies to $\frac{2y + 3x}{xy} \cdot \frac{xy}{1} = 2y + 3x$.
• Simplify $\frac{\frac{x}{y}}{\frac{x}{z}}$. This is a division problem: $\frac{x}{y} \div \frac{x}{z} = \frac{x}{y} \cdot \frac{z}{x} = \frac{z}{y}$.
• Simplify $\frac{2 + \frac{5}{a}}{\frac{a}{a+5}}$. Combine the numerator: $2 + \frac{5}{a} = \frac{2a+5}{a}$. Now the expression is $\frac{\frac{2a+5}{a}}{\frac{a}{a+5}}$, which equals $\frac{2a+5}{a} \cdot \frac{a+5}{a} = \frac{(2a+5)(a+5)}{a^2}$.