Factorising by Grouping: GCSE Maths Explained

When do you use factorising by grouping?

Use this method when an expression has four terms and no single factor divides all four. The classic sign is that the expression can be split into two pairs that share a common bracket after individual factorisation.

Common situations:

• Expressions of the form ax + ay + bx + by
• Some quadratics ax² + bx + cx + d arrived at by splitting the middle term (used in factorising quadratics where a ≠ 1)
• Expressions given explicitly in four-term form in GCSE exam questions

What is the step-by-step method?

1. Split the expression into two pairs of terms.
2. Factorise each pair by taking out the HCF of that pair.
3. Check that the bracket remaining after factorisation is identical in both pairs.
4. Write the final answer as the product of two brackets: the common bracket and the bracket formed by the two HCFs.

Worked example 1: straightforward grouping

Factorise 2x + 6 + xy + 3y.

Step 1: Group into two pairs: (2x + 6) + (xy + 3y).

Step 2: Factorise each pair:

• 2x + 6 = 2(x + 3)
• xy + 3y = y(x + 3)

Step 3: Both pairs contain the bracket (x + 3). Extract it:

• 2(x + 3) + y(x + 3) = (x + 3)(2 + y)

Answer: (x + 3)(y + 2) ✓

Verify by expanding: (x + 3)(y + 2) = xy + 2x + 3y + 6 = 2x + 6 + xy + 3y ✓

Worked example 2: grouping with subtraction

Factorise 6x² + 9x + 4x + 6.

Step 1: Group: (6x² + 9x) + (4x + 6).

Step 2: Factorise each pair:

• 6x² + 9x = 3x(2x + 3)
• 4x + 6 = 2(2x + 3)

Step 3: Common bracket is (2x + 3):

• 3x(2x + 3) + 2(2x + 3) = (2x + 3)(3x + 2)

Answer: (2x + 3)(3x + 2) ✓

Verify: (2x + 3)(3x + 2) = 6x² + 4x + 9x + 6 = 6x² + 13x + 6 ✓

Worked example 3: involving negative terms

Factorise x³ − 4x² + 3x − 12.

Step 1: Group: (x³ − 4x²) + (3x − 12).

Step 2: Factorise each pair:

• x³ − 4x² = x²(x − 4)
• 3x − 12 = 3(x − 4)

Step 3: Common bracket is (x − 4):

• x²(x − 4) + 3(x − 4) = (x − 4)(x² + 3)

Answer: (x − 4)(x² + 3) ✓

Note: (x² + 3) cannot be factorised further over the real numbers.

How is grouping used to factorise harder quadratics?

For ax² + bx + c where a ≠ 1, the splitting method uses grouping:

1. Find two numbers that multiply to a × c and add to b.
2. Rewrite the middle term as the sum of those two numbers times x.
3. Factorise by grouping.

Example: Factorise 6x² + 11x + 4.

• a × c = 6 × 4 = 24; find two numbers with product 24 and sum 11: 3 and 8.
• Rewrite: 6x² + 3x + 8x + 4
• Group: 3x(2x + 1) + 4(2x + 1) = (2x + 1)(3x + 4) ✓

What mistakes should you avoid?

• Splitting into unequal pairs. In example 2, splitting as (6x² + 4x) and (9x + 6) also works — try it — but make sure the common bracket that emerges is identical in both halves.
• Forgetting to check the common bracket matches. If the brackets after step 2 are different (e.g. (x + 2) and (x − 2)), you may need to re-order the four terms and try a different pairing.
• Leaving the answer as two factorised pairs. The final answer is the product of two brackets, not 3x(2x + 3) + 2(2x + 3). Extract the common bracket.
• Not verifying by expansion. Expanding your answer takes 30 seconds and will catch sign errors immediately.

Frequently asked questions

What if I can't find a common bracket after the first grouping?

Try re-ordering the four terms and grouping differently. For example, ax + bx + ay + by groups naturally as (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (a + b)(x + y). Sometimes simply rearranging gives the common bracket. If no rearrangement works, the expression may not be factorisable by grouping.

Can grouping factorise any four-term expression?

No. Factorising by grouping requires the expression to have a pair structure where a common bracket emerges. Expressions such as x³ + 2x² + 3x + 7 have no common bracket and cannot be factorised by grouping (or at all over the rationals). This method works when the four terms have the right relationships between their coefficients.

Is grouping the same as factorising out a common factor?

Not quite. Factorising out a common factor (or HCF) applies when one factor divides all the terms — giving a single bracket. Grouping applies when no single factor divides all four terms, but pairs of terms share a common factor. Grouping uses single-bracket factorisation twice, then combines the results.

Will factorising by grouping appear on both foundation and higher GCSE papers?

Factorising four-term expressions explicitly may appear on both tiers, but the splitting method for harder quadratics (where you generate the four-term form yourself) is more commonly tested on higher tier. Check your specification; for AQA GCSE Mathematics (8300), factorising quadratics ax² + bx + c is a higher-tier topic.

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