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Pure · Algebra & functions

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Algebraic fractions

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

Pure · Algebra & functions

Algebraic fractions

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

Why it works

Algebraic fractions obey the same rules as ordinary fractions — the only new skill is factorising first so you can see what cancels.

Simplifying. You may cancel a factor common to top and bottom, never a single term out of a sum. So factorise both fully, then cancel matching brackets: x2−9x2+x−6=(x−3)(x+3)(x−2)(x+3)=x−3x−2.\frac{x^2 - 9}{x^2 + x - 6} = \frac{(x-3)(x+3)}{(x-2)(x+3)} = \frac{x-3}{x-2}. The cancelled (x+3)(x+3) was a factor of each. You could not cancel the x2x^2 in x2+9x2\dfrac{x^2 + 9}{x^2} — there x2x^2 is only part of the sum on top.

Multiplying is straight across (top × top, bottom × bottom); dividing is "multiply by the reciprocal" — flip the second fraction: AB÷CD=AB×DC.\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}. Factorise everything first, then cancel before multiplying out.

Adding and subtracting needs a common denominator, exactly as with numbers: 2x+1+3x−2=2(x−2)+3(x+1)(x+1)(x−2)=5x−1(x+1)(x−2).\frac{2}{x+1} + \frac{3}{x-2} = \frac{2(x-2) + 3(x+1)}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)}. You cannot just add the tops and add the bottoms.