How to Divide Rational Expressions? (+FREE Worksheet!)

Dividing rational expressions uses the same “Keep-Change-Flip” (KCF) rule you learned when dividing ordinary fractions: keep the first expression, change division to multiplication, and flip the second expression. After that, the problem becomes multiplication of rational expressions — factor, cancel, and multiply. Mastering dividing rational expressions is a core Algebra 1 skill that carries into every algebra and precalculus course.

What Is Division of Rational Expressions?

Dividing \(\color{blue}{\frac{A}{B}}\) by \(\color{blue}{\frac{C}{D}}\) follows the same rule as regular fraction division:

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\(\color{blue}{(\frac{A}{B}) \div (\frac{C}{D}) = (\frac{A}{B}) \times (\frac{D}{C}) = \frac{(A \cdot D)}{(B \cdot C)}}\)

The divisor (the second fraction) is flipped to its reciprocal, and then you multiply. Factor and cancel before multiplying to keep the work simple.

How to Divide Rational Expressions

Step 1 — Keep-Change-Flip (KCF)

Keep the first fraction as it is. \(\color{blue}{\text{ Change } \div \text{ to } \times}\). Flip the second fraction (write its reciprocal).

Step 2 — Factor all polynomials

Factor every numerator and denominator completely.

Step 3 — Cancel and multiply

Cancel any factor that appears in a numerator and a denominator. Multiply remaining factors.

Example

Simplify \(\color{blue}{\frac{(x^{2} – 9)}{(x + 2)} \div \frac{(x – 3)}{(x + 2)}}\).

KCF: \(\color{blue}{\frac{(x^{2} – 9)}{(x + 2)} \times \frac{(x + 2)}{(x – 3)}}\)
Factor: \(\color{blue}{\frac{((x + 3)(x – 3))}{(x + 2)} \times \frac{(x + 2)}{(x – 3)}}\)
Cancel (\(\color{blue}{x + 2}\)) and (\(\color{blue}{x – 3}\)): \(\color{blue}{x + 3}\)  (x ≠ −2, x ≠ 3)

Step-by-Step Summary

1. Write the problem as \(\color{blue}{\frac{A}{B} \div \frac{C}{D}}\).
2. Apply KCF: rewrite as \(\color{blue}{\frac{A}{B} \times \frac{D}{C}}\).
3. Factor every numerator and denominator completely.
4. Cancel common factors across numerators and denominators.
5. Multiply remaining factors to get the simplified result.
6. State domain restrictions (all x-values making any denominator zero, including the flipped one).

Watch: Dividing Rational Expressions (Video Lesson)

The Organic Chemistry Tutor walks through the Keep-Change-Flip method for dividing rational expressions with multiple examples:

Dividing Rational Expressions – Worked Examples

Example 1: Simplify \(\color{blue}{\frac{(x^{2} – 9)}{(x + 2)} \div \frac{(x – 3)}{(x + 2)}}\).

KCF: \(\color{blue}{\frac{((x + 3)(x – 3))}{(x + 2)} \times \frac{(x + 2)}{(x – 3)}}\)
Cancel: \(\color{blue}{x + 3}\)  (x ≠ 3, x ≠ −2)

Example 2: Simplify 4x²/\(\color{blue}{(3y) \div \frac{8x}{(9y^{2})}}\).

KCF: \(\color{blue}{\frac{(4x^{2})}{(3y)} \times \frac{(9y^{2})}{(8x)}}\)
= \(\color{blue}{\frac{(4 \cdot 9 \cdot x^{2} \cdot y^{2})}{(3 \cdot 8 \cdot y \cdot x)}}\)
= \(\color{blue}{\frac{36x^{2}y^{2}}{(24\text{ xy })}}\)
Cancel: = \(\color{blue}{\frac{3\text{ xy }}{2}}\)

Example 3: Simplify \(\color{blue}{\frac{(x^{2} – 1)}{(x + 3)} \div \frac{(x – 1)}{(x + 3)}}\).

KCF: \(\color{blue}{\frac{((x + 1)(x – 1))}{(x + 3)} \times \frac{(x + 3)}{(x – 1)}}\)
Cancel (\(\color{blue}{x + 3}\)) and (\(\color{blue}{x – 1}\)): \(\color{blue}{x + 1}\)  (x ≠ −3, x ≠ 1)

Example 4: Simplify 6x²/\(\color{blue}{y \div \frac{3x}{y}}\)².

KCF: \(\color{blue}{(\frac{6x^{2}}{y}) \times (\frac{y^{2}}{3x})}\)
= \(\color{blue}{\frac{6x^{2}y^{2}}{(3\text{ xy })}}\) = 2xy  (y ≠ 0, x ≠ 0)

More Practice: Multiplying and Dividing Rational Expressions (Khan Academy)

Khan Academy covers both multiplying and dividing rational expressions in this Algebra II lesson with worked examples and interactive practice:

Exercises for Dividing Rational Expressions

1. \(\color{blue}{\frac{(x + 5)}{(x – 2)} \div \frac{(x + 5)}{(x + 3)}}\)
2. \(\color{blue}{\frac{(x^{2} – 4)}{(x + 1)} \div \frac{(x – 2)}{(x + 1)}}\)
3. 10x²/\(\color{blue}{(3y) \div \frac{5x}{(6y^{2})}}\)
4. \(\color{blue}{\frac{(x^{2} – 1)}{(x – 4)} \div \frac{(x + 1)}{(x – 4)}}\)
5. 8a²\(\color{blue}{\frac{b}{(c)} \div \frac{4a}{(c^{2})}}\)

Answers

1. KCF, cancel (\(\color{blue}{x + 5}\)): \(\color{blue}{\frac{(x + 3)}{(x – 2)}}\)
2. \(\color{blue}{\frac{((x + 2)(x – 2))}{(x + 1)} \times \frac{(x + 1)}{(x – 2)}}\); cancel (\(\color{blue}{x – 2}\)) and (\(\color{blue}{x + 1}\)): \(\color{blue}{x + 2}\)
3. \(\color{blue}{\frac{(10x^{2})}{(3y)} \times \frac{(6y^{2})}{(5x)} = 60x}\)\(\color{blue}{^{2}\frac{y^{2}}{(15\text{ xy })}}\) = 4xy
4. \(\color{blue}{\frac{((x + 1)(x – 1))}{(x – 4)} \times \frac{(x – 4)}{(x + 1)}}\); cancel (\(\color{blue}{x + 1}\)) and (\(\color{blue}{x – 4}\)): \(\color{blue}{x – 1}\)
5. \(\color{blue}{(\frac{8a^{2}b}{c}) \times (\frac{c^{2}}{4a}) = 8a}\)\(\color{blue}{^{2}\frac{\text{ bc }^{2}}{(4\text{ ac })}}\) = 2abc

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Want More Practice?

We haven’t published a worksheet built specifically for Dividing Rational Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work. They’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Multiplying Polynomials Worksheet
• Download Factoring Trinomials x2 bx c Worksheet

Frequently Asked Questions

What is dividing rational expressions?

Dividing rational expressions means dividing one algebraic fraction by another, such as (x + 1)/x ÷ (x + 1)/x². It works just like dividing number fractions: you multiply the first expression by the reciprocal of the second. The difference is that you factor polynomials so you can cancel common factors, and you must track every value of x that would make a denominator zero.

How do you divide rational expressions step by step?

First, keep the first expression, change ÷ to ×, and flip the second expression. Second, factor every numerator and denominator. Third, cancel factors that appear in both a numerator and a denominator. Fourth, multiply what is left and list the restrictions. Example: (x² − 9)/(x + 2) ÷ (x − 3)/(x + 2) becomes (x + 3)(x − 3)/(x + 2) × (x + 2)/(x − 3) = x + 3, with x ≠ −2 and x ≠ 3.

How is dividing rational expressions different than multiplying?

Dividing adds one step before multiplying: you flip the second expression to its reciprocal. After that, factoring, canceling and multiplying are the same. Flipping also adds a restriction, because the old numerator of the divisor becomes a denominator. In ÷ (x − 3)/(x + 2), x cannot be −2 or 3, while multiplying by that same expression would only rule out −2.

What is an example of dividing rational expressions?

Try 2x/(x + 1) ÷ 4x²/(x + 1). Flip the second fraction and multiply: 2x/(x + 1) × (x + 1)/(4x²). The (x + 1) factors cancel, leaving 2x/(4x²). Divide out 2x to get 1/(2x). Because x + 1 and 4x² both appeared as denominators along the way, the answer holds for x ≠ 0 and x ≠ −1.

How do you divide rational expressions involving multivariate monomials?

Flip the second fraction, multiply, and then simplify each variable separately using exponent rules. For 6x²y/(5z) ÷ 3xy²/(10z²), multiply 6x²y/(5z) by 10z²/(3xy²) to get 60x²yz²/(15xy²z). Then 60 ÷ 15 = 4, x²/x = x, y/y² = 1/y and z²/z = z, so the answer is 4xz/y, with x, y and z not equal to 0.

Related Topics

• Multiplying Rational Expressions
• Adding and Subtracting Rational Expressions
• Simplifying Complex Fractions
• Graphing Rational Functions
• Multiplying and Dividing Functions