Algebraic Fractions: Simplify, Add & Subtract (Grade 10)

Simplifying algebraic fractions by factorising, multiplying and dividing them, and adding or subtracting using a lowest common denominator — including denominators that factorise as cubes.

Algebraic fractions behave exactly like ordinary fractions. The one extra step is that you must factorise first — nothing can be cancelled until the top and bottom are products.

1Simplifying: factorise, then cancel FACTORS

Worked ExampleSimplify: x² − 9x + 3

1
Factorise the top: x² − 9 = (x − 3)(x + 3).
A difference of two squares.
2
(x − 3)(x + 3)(x + 3)
Now top and bottom are products.
3
Cancel the (x + 3): answer x − 3.

⚠️

You may only cancel factors, never single terms. In x + 33 the 3s do not cancel, because the top is a sum, not a product.

Worked ExampleSimplify: x² + 5x + 6x² − 4

1
Top: x² + 5x + 6 = (x + 2)(x + 3).
Trinomial.
2
Bottom: x² − 4 = (x − 2)(x + 2).
Difference of squares.
3
Cancel (x + 2): x + 3x − 2

2Multiplying and dividing

Multiply: factorise everything, cancel across, then multiply what is left. Divide: flip the second fraction and multiply.

Worked ExampleSimplify: x² − 1x ÷ x + 13x

1
Flip and multiply: x² − 1x × 3xx + 1.
Dividing by a fraction is multiplying by its reciprocal.
2
Factorise: x² − 1 = (x − 1)(x + 1).
3
Cancel (x + 1) and one x: 3(x − 1).

3Adding and subtracting

The four steps for adding algebraic fractions. Step 1 is the one learners skip.

Worked ExampleSimplify: 2x + 1 + 3x − 1

1
The denominators do not factorise further, so the LCD is (x + 1)(x − 1).
2
Rewrite: 2(x − 1)(x + 1)(x − 1) + 3(x + 1)(x + 1)(x − 1).
Multiply each fraction by what its denominator is missing.
3
Top: 2x − 2 + 3x + 3 = 5x + 1.
Expand and collect.
4
= 5x + 1x² − 1
The denominator may be left factorised or expanded.

⚠️

When subtracting, the minus applies to the whole numerator. ad − b + cd = a − b − cd — bracket the second numerator before you subtract.

4When a denominator is a sum or difference of cubes

Worked ExampleSimplify: x − 2x³ − 8

1
Factorise the bottom: x³ − 8 = (x − 2)(x² + 2x + 4).
A difference of two cubes.
2
(x − 2)(x − 2)(x² + 2x + 4)
3
Cancel (x − 2): 1x² + 2x + 4
The numerator becomes 1, not 0 — a very common slip.

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Whenever you see a cube in a denominator, expect it to factorise and cancel. That is exactly why the cube identities are taught just before this topic.