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.
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
Avoid These
The most common mistakes students make.
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 parents and students ask.
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