Procedure

Combine algebraic fractions over one denominator

Algebra • Expressions • Simplification

Statement

To combine a sum or difference of algebraic fractions into one fraction: factorise every denominator, take the LCM of those factorisations as the common denominator, multiply each numerator by the factors its own denominator is missing, add and subtract the numerators over that single denominator, and finally cancel any factor common to the combined numerator and the denominator. The values making any original denominator zero are excluded throughout, and the identity holds for all other values.

How to use it

1Factorise and simplify each fraction on its own first; a fraction that cancels down shortens the common denominator everyone else has to carry.
2Build the common denominator from the factorisations, not by multiplying the denominators together - the product is a common denominator but usually a needlessly large one.
3Keep the numerator unexpanded in brackets until every term is in place, then expand and collect once; expanding early is where signs are lost.
4Expect the numerator to factorise against the denominator at the end - in a question built to have a tidy answer, it will.

Worked example

Simplify

\frac{1}{x} + \frac{1}{x + 1} - \frac{1}{x ( x + 1 )}

Step-by-step

1
Common denominator $= x (x + 1)$
Why: The third denominator is already the LCM of the other two.
2
$= \frac{( x + 1 ) + x - 1}{x ( x + 1 )}$
Why: The first numerator is short of $(x + 1)$ and the second short of $x$ .
3
$= \frac{2 x}{x ( x + 1 )} = \frac{2}{x + 1}$
Why: Collect the numerator, then cancel the common factor $x$ .

Answer

\frac{2}{x + 1}

Watch out

This is an expression, not an equation: the denominator cannot be multiplied away, only cancelled against a matching factor of the numerator. Subtracting a fraction subtracts its whole numerator, so the bracket must be kept -

- \frac{a - c}{d}

contributes

- a + c

, not

- a - c

. And the cancellation at the end is only valid away from the excluded values.

Builds on

LCM and HCF of several numbers from prime powers

Also known as

add algebraic fractions, combine fractions over an LCM, single fraction, common denominator for an expression, simplify a sum of rational expressions

Appears in past papers

2020 (II)

Q34