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\(\color{blue}{\frac{b}{(c)} \div \frac{4a}{(c^{2})}}\)

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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

Why do I flip only the second fraction?

Division by a fraction is the same as multiplication by its reciprocal: \(\color{blue}{a \div (\frac{c}{d}) = a \times (\frac{d}{c})}\). You flip only the divisor (the fraction doing the dividing), not the dividend (the fraction being divided).

What domain restrictions apply to division?

You must exclude any x-value that makes any original denominator zero — including the denominator of the second fraction before it is flipped. After flipping, that denominator becomes a numerator, but the restriction still applies.

What is the difference between multiplying and dividing rational expressions?

Multiplying: multiply numerator by numerator and denominator by denominator, then simplify. Dividing: flip the second expression to its reciprocal and then follow the same multiplication procedure. The only extra step in division is the flip.

Related Topics

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