Algebraic Fractions: AQA Level 2 Further Maths revision notes

Cancel factors and record restrictions

A denominator cannot be zero. Record excluded values from the original expression before cancelling. Simplified expressions agree with the original only on its allowed inputs.

Cancel a factor multiplying the whole numerator and denominator. Do not cancel terms across an addition: \frac{x+3}{x} is not 3. Factorising first makes valid cancellations visible.

Worked example 1

Simplify \frac{x^2-9}{x^2+x-6} and state its excluded values.

Factorise: \frac{(x-3)(x+3)}{(x+3)(x-2)}. The original denominator is zero at x=-3 or x=2
Your turn. Which common factor can be cancelled?

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Cancel x+3 on the allowed domain.
Answer: \frac{x-3}{x-2}, with x\ne-3,2. Cancelling does not make x=-3 an allowed input.

Explore a cancelled factor

Try it: set x=2. The simplified line would give 4, but the original fraction has denominator zero, so the point (2,4) is missing.

Add and subtract using a common denominator

Factorise the denominators to find a useful common denominator. Multiply the top and bottom of each fraction by the same missing factor. When subtracting, put the whole second numerator in brackets.

Worked example 2

Write \frac{2}{x}+\frac{3}{x+1} as a single fraction.

Use denominator x(x+1): \frac{2(x+1)+3x}{x(x+1)}
Your turn. Simplify 2(x+1)+3x

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

The numerator is 5x+2
Answer: \frac{5x+2}{x(x+1)}, with x\ne0,-1

Worked example 3

Write \frac{x+3}{x-1}-\frac{x-2}{x+2} as a single simplified fraction.

Use denominator (x-1)(x+2). The numerator is (x+3)(x+2)-(x-2)(x-1)
Expand the numerator: (x^2+5x+6)-(x^2-3x+2)
Your turn. Simplify this numerator.

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

x^2+5x+6-x^2+3x-2=8x+4
Answer: \frac{8x+4}{(x-1)(x+2)}, with x\ne1,-2. There is no common factor to cancel.

Multiply fractions; divide by a reciprocal

For multiplication, factorise and cancel common factors before multiplying what remains. For division, multiply by the reciprocal of the second fraction. The divisor must be defined and must not be zero.

Worked example 4

Simplify \frac{x^2-4}{3x}\times\frac{6x^2}{x+2}

Factorise the difference of squares: \frac{(x-2)(x+2)\,6x^2}{3x(x+2)}. Initially x\ne0,-2
Your turn. After cancelling, what is \frac{6x^2}{3x}?

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

\frac{6x^2}{3x}=2x for x\ne0
Answer: 2x(x-2), with x\ne0,-2

Worked example 5

Simplify \frac{x^2-1}{x+3}\div\frac{x-1}{x^2-9}, stating all excluded values.

The original denominators exclude x=-3,3. The divisor is zero at x=1, so this is excluded too.
Multiply by the reciprocal: \frac{(x-1)(x+1)}{x+3}\times\frac{(x-3)(x+3)}{x-1}
Your turn. After cancelling common factors, what remains?

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

The factors left are (x+1)(x-3)
Answer: (x+1)(x-3), with x\ne-3,1,3. Keep all three restrictions.

Worked example 6

Solve \frac{x}{x-3}=\frac{5x-6}{x^2-3x}

Factorise the denominator as x(x-3), so x\ne0,3. Multiply both sides by x(x-3) to obtain x^2=5x-6
Rearrange and factorise: x^2-5x+6=(x-2)(x-3)=0. The candidates are 2 and 3
Your turn. Which candidate is allowed in the original equation?

Your answer

x=2 is allowed; x=3 gives a zero denominator.
Answer: x=2. Substitution gives -2 on both sides.

Common mistakes

Cancelling terms. Factorise first; cancel common factors of the whole numerator and denominator.
Losing excluded values. Keep every restriction from the original expression after cancelling.
Adding or subtracting denominators. Use a common denominator, then combine the numerators.
Losing a subtraction sign. Bracket the whole second numerator before subtracting it.
Dividing by zero. Check the divisor is defined and non-zero before taking its reciprocal.