The idea

Algebraic fractions

Simplifying, multiplying, dividing, adding and subtracting fractions with polynomials — by factorising and cancelling only common factors.

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Pure · Algebra & functions

Algebraic fractions

Simplifying, multiplying, dividing, adding and subtracting fractions with polynomials — by factorising and cancelling only common factors.

Why it works

The cancel that costs the mark

Simplify

\dfrac{x^2 - 9}{x^2 + x - 6}

. The tempting move — crossing out the

x^2

on top against the

x^2

below — scores zero, and it is one of the most-penalised slips in the subject. Algebraic fractions obey exactly the same rules as ordinary fractions, and the governing law is: you may cancel a factor shared by the whole top and the whole bottom — never a single term out of a sum. The only new skill is factorising first, so the shared factor becomes visible:

\frac{x^{2} - 9}{x^{2} + x - 6} \;=\; \frac{(x-3)(x+3)}{(x-2)(x+3)} \;=\; \frac{x-3}{x-2}

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The rest of the explanation, plus 3 worked examples you step through move by move.

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