Algebra
Grade 7-9Algebraic Fractions
Algebraic fractions follow exactly the same rules as numerical fractions, but factorising is usually the key first step that makes simplifying, combining or solving them possible. This lesson covers simplifying an algebraic fraction by factorising, multiplying and dividing algebraic fractions, adding and subtracting algebraic fractions, and solving an equation that contains algebraic fractions.
What you need to know
- Factorise the top and bottom first, then cancel common brackets.
- To add or subtract, use a common denominator, usually the product of the two denominators.
- To divide, flip the second fraction and multiply.
- You can only cancel whole factors, never individual terms.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Simplifying an algebraic fraction by factorising
Factorise the numerator and the denominator fully, then cancel any factor that appears in both.
(x² − 9) ÷ (x + 3) has a numerator that factorises as (x + 3)(x − 3), so the fraction becomes [(x + 3)(x − 3)] ÷ (x + 3), and cancelling (x + 3) leaves x − 3.
Multiplying and dividing algebraic fractions
To multiply, multiply the numerators together and the denominators together, simplifying where possible. To divide, multiply by the reciprocal of the second fraction, flipping its numerator and denominator.
(x/3) × (6 ÷ x²) means (x/3) × (6/x²), which multiplies to give 6x/(3x²), simplifying to 2/x.
Adding and subtracting algebraic fractions
Find a common denominator by multiplying the two denominators together, rewrite each fraction over this common denominator, then add or subtract the numerators.
3/x + 2/(x + 1) has a common denominator of x(x + 1). Rewriting gives [3(x + 1) + 2x] ÷ [x(x + 1)], which simplifies to (5x + 3) ÷ [x(x + 1)].
Solving an equation containing algebraic fractions
Multiply every term by the common denominator to clear the fractions, then solve the resulting equation, which is often a quadratic.
To solve 1/x + 4/(x + 3) = 1, multiply through by x(x + 3) to get (x + 3) + 4x = x(x + 3), which simplifies to x² − 2x − 3 = 0. Factorising gives (x − 3)(x + 1) = 0, so x = 3 or x = −1.
Worked examples
Three exam-style questions, fully solved
Simplify (x² − 9) ÷ (x + 3).
Easy- 1.Factorise the numerator as a difference of two squares: (x + 3)(x − 3)
- 2.Cancel the common factor of (x + 3)
Answer: x − 3
Write 3/x + 2/(x + 1) as a single fraction.
Medium- 1.Find the common denominator: x(x + 1)
- 2.Rewrite each fraction over this denominator and combine the numerators: [3(x + 1) + 2x] ÷ [x(x + 1)]
- 3.Simplify the numerator: 3x + 3 + 2x = 5x + 3
Answer: (5x + 3) ÷ [x(x + 1)]
Solve 1/x + 4/(x + 3) = 1.
Hard- 1.Multiply every term by the common denominator, x(x + 3): (x + 3) + 4x = x(x + 3)
- 2.Simplify and rearrange into a quadratic: 5x + 3 = x² + 3x, so x² − 2x − 3 = 0
- 3.Factorise and solve: (x − 3)(x + 1) = 0
Answer: x = 3 or x = −1
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Simplify (x² − 9) ÷ (x + 3).
Write 3/x + 2/x as a single fraction.
Write x/2 + x/3 as a single fraction.
Simplify (2x + 6) ÷ (x² + 3x).
Simplify (x² + 5x + 6) ÷ (x² + 2x).
Solve 6/x = 3.
Avoid these
The mistakes students make most often
Cancelling a term from a fraction without factorising first, such as cancelling an x that is added rather than multiplied throughout the numerator or denominator.
When dividing algebraic fractions, forgetting to take the reciprocal of the second fraction before multiplying.
When adding or subtracting algebraic fractions, forgetting to multiply each numerator by the same factor used to reach the common denominator.
When solving an equation with algebraic fractions, forgetting to multiply every single term by the common denominator, not just the fractions.
After solving a quadratic that results from clearing the fractions, forgetting to check that a solution does not make one of the original denominators zero, which would make it invalid.
FAQ
Questions students and parents ask
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What to learn next
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