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Algebra · Algebraic fractions

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Simplifying, multiplying and dividing algebraic fractions

Why only FACTORS cancel — never terms — so factorising must come first, and how × and ÷ work exactly as for number fractions once everything is in factors.

Algebra · Algebraic fractions

Simplifying, multiplying and dividing algebraic fractions

Why only FACTORS cancel — never terms — so factorising must come first, and how × and ÷ work exactly as for number fractions once everything is in factors.

Why it works

An algebraic fraction obeys every rule a number fraction does — and one above all: cancelling divides the whole top and the whole bottom, so only common FACTORS cancel. A term glued into a sum cannot be cancelled alone: in x2+5xx2−25\frac{x^2 + 5x}{x^2 - 25}, striking out the x2x^2's is illegal (they're terms, not factors — test x=1x = 1: 6−24\frac{6}{-24}, but "cancelling" gives 5−25\frac{5}{-25}; different, so wrong).

The routine is always: FACTORISE TOP, FACTORISE BOTTOM, THEN cancel.

x2+5xx2−25=x(x+5)(x+5)(x−5)=xx−5.\frac{x^2 + 5x}{x^2 - 25} = \frac{x(x + 5)}{(x + 5)(x - 5)} = \frac{x}{x - 5}.

The whole bracket (x+5)(x + 5) cancels because it's a factor of both. Every Higher simplification is this move with different factorising inside — quadratics included:

x2+7x+12x2−9=(x+3)(x+4)(x+3)(x−3)=x+4x−3,\frac{x^2 + 7x + 12}{x^2 - 9} = \frac{(x + 3)(x + 4)}{(x + 3)(x - 3)} = \frac{x + 4}{x - 3},

and with a leading coefficient, 2x2+x−3x2−1=(2x+3)(x−1)(x+1)(x−1)=2x+3x+1\frac{2x^2 + x - 3}{x^2 - 1} = \frac{(2x + 3)(x - 1)}{(x + 1)(x - 1)} = \frac{2x + 3}{x + 1}. Cancel whole brackets only — never a lone xx out of (x+4)(x + 4), never the 3's in 2x+3x−3\frac{2x + 3}{x - 3}.

Multiplying: tops together, bottoms together — after factorising.

x+23×6x2+2x=x+23×6x(x+2)=2x\frac{x + 2}{3} \times \frac{6}{x^2 + 2x} = \frac{x + 2}{3} \times \frac{6}{x(x + 2)} = \frac{2}{x}

— the (x+2)(x + 2)'s cancel across the multiplication, and 6/3=26/3 = 2. No common denominator is ever needed for ×\times.

Dividing: keep the first, flip the second — then it's a multiply:

x2−45÷x+210=(x−2)(x+2)5×10x+2=2(x−2).\frac{x^2 - 4}{5} \div \frac{x + 2}{10} = \frac{(x - 2)(x + 2)}{5} \times \frac{10}{x + 2} = 2(x - 2).

Sanity-check with a number. Substitute an easy xx into the original and your answer; disagreement means an illegal cancel somewhere.