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Algebraic fractions worksheets

You may cancel a factor and you may not cancel a term, and (x² − 81) ÷ (x − 9) is where the difference bites. These free printable worksheets show the illegal cancellation written out and ask which answer is correct.

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Algebraic fractions
Algebraic fractions worksheet — Cancelling a factor, not a term, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Factorising first is what makes cancelling legal, and the answer key shows that step rather than asserting the rule. Once the top is written as two brackets, the one that matches the bottom leaves and the other one is the answer — and which one leaves depends on the sign underneath, so the answer is not always the same shape.

Why this sheet works

Adding two algebraic fractions is the same discipline from the other end. The denominators have to be made common rather than added, and the four wrong answers are the four shortcuts: bottoms added, tops and bottoms both added, tops multiplied, and the sum used as the common denominator.

How this one works

One question from this sheet, worked through a step at a time.

Mia wrote (x² − 81) ÷ (x − 9) − x = x + 9. Which is correct?

(x² − 81) ÷ (x − 9) − x
(x² − 81) ÷ (x − 9) − x=(x + 9) − x
(x² − 81) ÷ (x − 9) − x=(x + 9) − x
(x² − 81) ÷ (x − 9) − x=9
(x² − 81) ÷ (x − 9) − x=9
they wrote:x + 9
  1. Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
  2. On your own, and writing the step nobody writes down: (x² − 81) ÷ (x − 9) − x becomes (x + 9) − x.
  3. Tidy that up and it is 9.
  4. Now compare. They wrote x + 9, and the x was never taken away at the end — so the correct answer is 9.

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