Collecting terms and expanding brackets
Use the distributive law to turn an expression into a sum of comparable terms.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Multiply signed numbers.
- Read coefficients and powers in an expression.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which term can be collected with 4x²?
What you will learn
- Collect terms with the same variable part.
- Expand a single bracket and two brackets.
- Factorise by extracting the greatest common factor.
Like terms share their entire variable part
The terms 3x² and −5x² are alike because both count x². They combine to −2x². The term 4x cannot join them: x and x² are different quantities. For 3a + 2b − a + 5b, collect the a terms and b terms separately to obtain 2a + 7b.
A sign belongs to the term that follows it. In 7x − 2y − 3x + y, the y terms are −2y and +y, giving −y. Reordering terms is useful only if their signs move with them. A missing written coefficient in x means one, while −x means negative one.
Distribute to every term
The product 3(2x − 5) equals 6x − 15 because three multiplies both parts. With a negative multiplier, −2(x − 4) = −2x + 8. For an algebraic multiplier, 4x(2x + y) = 8x² + 4xy. The power x² appears because x multiplies another x.
To expand (x + 2)(x − 3), multiply each term of the first bracket by each term of the second: x² − 3x + 2x − 6. Then collect to give x² − x − 6. There are four products before collecting; overlooking the two middle products is a common source of error.
Reverse expansion to extract a common factor
Factorising writes a sum as a product. For 12x² + 18xy, both terms contain 6x, so the expression is 6x(2x + 3y). Divide each original term by the extracted factor to find what remains. Choose the greatest shared numerical factor and the minimum shared variable powers so no common factor is left inside.
Expansion checks factorisation: 6x multiplied by 2x and by 3y restores both original terms. For 8a − 12, the greatest common factor is four, giving 4(2a − 3). Factorisation does not find an unknown value by itself. There must be an equation before you can solve for a letter.
Check by structure and substitution
Squaring a binomial means multiplying two identical brackets: (x + 4)² = x² + 8x + 16. It is not x² + 16. For (2x − 3)(x + 5), the middle terms are 10x and −3x, so the result is 2x² + 7x − 15.
An expansion should have the expected leading power and constant. Substituting x = 0 checks the constant; another input can reveal a missing cross term. A numerical check can find an error but is not a proof of an identity. Distributing to all terms is the proof. Extended work also expands several brackets or brackets containing different variables by the same rule.
Pause and explain
What is −3(2x − 1)?
Worked example
Expand and simplify (3x − 2)(x + 4) − 2x(x − 1).
Show the worked solution
- The first product is 3x² + 12x − 2x − 8 = 3x² + 10x − 8.
- The second product is 2x² − 2x; subtract the entire expression.
- 3x² + 10x − 8 − 2x² + 2x = x² + 12x − 8.
Answer x² + 12x − 8
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Simplify 5a − 3b + 3a − b, then factorise fully.
Give me a hint
Keep each sign attached when grouping terms, then look for a common factor.
Compare my reasoning
- Group 5a + 3a and −3b − b to obtain 8a − 4b.
- The greatest common numerical factor is four.
- Write 4(2a − b); expanding checks the simplified expression.
4(2a − b)
Look for these in your work
- Grouped only matching variable parts.
- Extracted the greatest common factor.
Expand (2x + 3)(x − 4).
Give me a hint
Write all four products before simplifying.
Compare my reasoning
- The products are 2x², −8x, 3x and −12.
- Combine −8x + 3x = −5x.
- The expression is 2x² − 5x − 12; at x = 0 both forms give −12.
2x² − 5x − 12
Look for these in your work
- Included both cross products.
- Checked the constant term.
A rectangle has sides (x + 3) cm and (x + 7) cm. Express its area as a polynomial and find the area for x = 2.
Give me a hint
Area multiplies the complete side expressions.
Compare my reasoning
- A = (x + 3)(x + 7).
- Expand to A = x² + 10x + 21.
- At x = 2, A = 4 + 20 + 21 = 45 cm², agreeing with 5 × 9.
A = x² + 10x + 21; 45 cm²
Look for these in your work
- Multiplied the side lengths rather than adding them.
- Checked against the actual rectangle.
Common mistakes
- Combining x and x².
- Dropping a term's negative sign when regrouping.
- Squaring each binomial term without the cross products.
What can you explain now?
Expand (x − 5)².
Compare with the explanation
x² − 10x + 25
Multiply two copies of (x − 5); the two middle products total −10x.
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Make it stick
Explain the four products in (a + b)(c + d) without using a memorised acronym. Check an expansion at two chosen values.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.