Extended. Algebraic fractions follow exactly the same rules as number fractions. Factorise and cancel to simplify, flip and multiply to divide, and use a common denominator to add and subtract.
Only cancel factors, never terms. Factorise the top and bottom completely first; then a bracket that appears on both can be cancelled. To add or subtract, write both fractions over a common denominator.
Section 01
Factorise, then cancel.
In the lab choose Cancel and press Next step. Watch the top and bottom be factorised fully before anything is crossed out, and notice that whole brackets cancel, never single terms. Press New question and predict what will cancel.
Factors are things multiplied together. Only a factor of the whole top AND the whole bottom can cancel.
\(\dfrac{x + 3}{x}\) does not simplify: the \(x\) on top is a term (added), not a factor.
\(\dfrac{x(x + 3)}{x}\) simplifies to \(x + 3\): here \(x\) multiplies the whole top.
Worked example Cancel a common bracket
Simplify \(\dfrac{x^{2} - 2x}{x^{2} - 5x + 6}\).
Top: common factor \(x\), so \(x^{2} - 2x = x(x - 2)\).
Bottom: product 6, sum \(-5\), so \(x^{2} - 5x + 6 = (x - 2)(x - 3)\).
\(\dfrac{x(x - 2)}{(x - 2)(x - 3)}\): cancel the common factor \((x - 2)\).
Answer: \(\dfrac{x}{x - 3}\)
Worked example Difference of two squares underneath
Cancelling the \(x^{2}\) terms in \(\dfrac{x^{2} + 3x}{x^{2} - 9}\) to get \(\dfrac{3x}{-9}\) is wrong: \(x^{2}\) is a term, not a factor. Factorise first: the answer is \(\dfrac{x}{x - 3}\).
Section 02
Multiply and divide.
Multiply tops together and bottoms together, cancelling common factors on the way. To divide, multiply by the reciprocal of the second fraction (flip it).
Find a common denominator, rewrite each fraction over it, then combine the numerators. Put each numerator in brackets so a minus sign reaches every term.
Worked example Numerical denominators
Write \(\dfrac{x}{3} + \dfrac{x - 4}{2}\) as a single fraction.
Common denominator 6: \(\dfrac{2x}{6} + \dfrac{3(x - 4)}{6}\).
Numerator: \(2x + 3x - 12 = 5x - 12\).
Answer: \(\dfrac{5x - 12}{6}\)
Worked example Subtracting a bracket
Write \(\dfrac{2x}{3} - \dfrac{3(x - 5)}{2}\) as a single fraction.
Common denominator 6: \(\dfrac{4x}{6} - \dfrac{9(x - 5)}{6}\).
\(-9(x - 5) = -9x + 45\).
Numerator: \(4x - 9x + 45 = 45 - 5x\).
Answer: \(\dfrac{45 - 5x}{6}\)
Worked example Algebraic denominators
Write \(\dfrac{1}{x - 2} + \dfrac{x + 1}{x - 3}\) as a single fraction.
Common denominator \((x - 2)(x - 3)\).
Numerator: \(1(x - 3) + (x + 1)(x - 2)\).
\((x + 1)(x - 2) = x^{2} - x - 2\), so the numerator is \(x - 3 + x^{2} - x - 2 = x^{2} - 5\).
Answer: \(\dfrac{x^{2} - 5}{(x - 2)(x - 3)}\)
Your turn
Write \(\dfrac{2}{x + 3} + \dfrac{5}{x - 4}\) as a single fraction in its simplest form.
In \(\dfrac{x + 1}{4} - \dfrac{x - 3}{6}\) the minus applies to the whole of \(x - 3\): \(3(x + 1) - 2(x - 3) = 3x + 3 - 2x + 6\). Writing \(-6\) instead of \(+6\) is the classic slip.
Stage 2 · Exam
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Simplify \(\dfrac{x^{2} + 3x}{x^{2} - 9}\).
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Write \(\dfrac{x + 1}{4} - \dfrac{x - 3}{6}\) as a single fraction in its simplest form.
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Simplify \(\dfrac{2x^{2} + 5x - 3}{4x^{2} - 1}\).
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Write \(\dfrac{3}{x + 2} - \dfrac{2}{x - 1}\) as a single fraction.
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