Further Expansion

Expanding a term times two brackets: expand the brackets first, then multiply every term, with checks for signs and squared brackets.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 26 real past-paper questions below

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A Term × Two Brackets (Extended/Higher)

Some questions multiply a term like 2x by two brackets, such as $2 x (x + 4) (3 x - 1)$ . Expand the two brackets first, then multiply by the term.

Rule

Two brackets first, then the term in front multiplies every term.

Key idea

Show every line. Expanding the two brackets correctly earns a mark on its own, so even if the last line slips you keep marks for the working.

Brackets First, Then the Term

Keep the term in front waiting until the brackets are done.

Worked example

1
$2 x (x + 4) (3 x - 1)$ — The question. 2x waits in front while we work on the two brackets.
2
$(x + 4) (3 x - 1) = 3 x^{2} - x + 12 x - 4$ — Multiply each term in the first bracket by each term in the second: x × 3x, x × (−1), 4 × 3x, 4 × (−1). Four products.
3
$= 3 x^{2} + 11 x - 4$ — Collect the x terms: −x + 12x = 11x. The brackets are done.
4
$2 x (3 x^{2} + 11 x - 4)$ — Now bring in the term in front. It must multiply all three terms.
5
$= 6 x^{3} + 22 x^{2} - 8 x$ — 2x × 3x² = 6x³, 2x × 11x = 22x², 2x × (−4) = −8x. The powers add: x × x² = x³.

Common mistake

Multiplying 2x into both brackets, like (2x² + 8x)(6x² − 2x). That multiplies by 2x twice and gives an x⁴ term.

Key idea

Any pair can go first. $2 x (x + 4) = 2 x^{2} + 8 x$ , then times (3x − 1), gives the same answer. Pick the pair that looks easiest.

Bigger Numbers in the Brackets

With numbers in front of x, multiply the numbers carefully. The method does not change.

Worked example

1
$5 x (2 x + 3) (4 x - 1)$ — The question. Same plan: brackets first, 5x waits.
2
$(2 x + 3) (4 x - 1) = 8 x^{2} - 2 x + 12 x - 3$ — Four products. Multiply numbers and letters: 2x × 4x = 8x², 2x × (−1) = −2x, 3 × 4x = 12x, 3 × (−1) = −3.
3
$= 8 x^{2} + 10 x - 3$ — Collect the x terms: −2x + 12x = 10x.
4
$5 x (8 x^{2} + 10 x - 3)$ — Now the term in front multiplies every term.
5
$= 40 x^{3} + 50 x^{2} - 15 x$ — Numbers times numbers, x times x: 5 × 8 = 40 with x³, 5 × 10 = 50 with x², 5 × (−3) = −15 with x.

Two Minus Signs

Minus times minus is plus. Watch the last term of the brackets.

Worked example

1
$4 x (x - 6) (x - 2)$ — The question. Both brackets hold a minus, so the last product needs care.
2
$(x - 6) (x - 2) = x^{2} - 2 x - 6 x + 12$ — Four products. The last one is (−6) × (−2) = +12, because minus times minus is plus.
3
$= x^{2} - 8 x + 12$ — Collect: −2x − 6x = −8x. Two minus terms added stay minus.
4
$4 x (x^{2} - 8 x + 12)$ — Now the term in front multiplies every term.
5
$= 4 x^{3} - 32 x^{2} + 48 x$ — 4x × x² = 4x³, 4x × (−8x) = −32x², 4x × 12 = +48x.

Key idea

Check with x = 1. The question gives 4 × (−5) × (−1) = 20, and your answer gives 4 − 32 + 48 = 20. They match.

A Squared Bracket (Extended/Higher)

A squared bracket means the bracket times itself. Write it out twice, so you cannot lose the middle term.

Worked example

1
$2 x (3 x - 4)^{2}$ — The question. The square belongs to the bracket only, not to the 2x.
2
$(3 x - 4)^{2} = (3 x - 4) (3 x - 4)$ — Write the bracket twice. Now it is an ordinary pair of brackets.
3
$= 9 x^{2} - 12 x - 12 x + 16$ — Four products. The last one is (−4) × (−4) = +16.
4
$= 9 x^{2} - 24 x + 16$ — Collect: −12x − 12x = −24x. This middle term is the one people forget.
5
$2 x (9 x^{2} - 24 x + 16)$ — Now the term in front multiplies every term.
6
$= 18 x^{3} - 48 x^{2} + 32 x$ — 2x × 9x² = 18x³, 2x × (−24x) = −48x², 2x × 16 = +32x.

Common mistake

Writing (3x − 4)² = 9x² + 16. Squaring each part on its own loses the −24x in the middle.

Show It in a Form (Extended/Higher)

Some questions give the shape of the answer: “show that … can be written as $a y^{3} + b y^{2} + cy$ ”. You already know the answer, so the marks are for the lines you write.

Worked example

1
$2 y (y + 6) (3 y - 2)$ — The question, with y instead of x. The letter changes nothing.
2
$(y + 6) (3 y - 2) = 3 y^{2} - 2 y + 18 y - 12$ — Expand the two brackets first: four products.
3
$= 3 y^{2} + 16 y - 12$ — Collect: −2y + 18y = 16y.
4
$2 y (3 y^{2} + 16 y - 12)$ — Now the term in front multiplies every term.
5
$= 6 y^{3} + 32 y^{2} - 24 y$ — 2y × 3y² = 6y³, 2y × 16y = 32y², 2y × (−12) = −24y.
6
$a = 6, b = 32, c = - 24$ — Match each term to $a y^{3} + b y^{2} + cy$ . Keep the minus with c: it is −24, not 24.

Key idea

In a “show that”, never jump straight to the given answer. Write each line, so the examiner sees where every term came from.

Which Method? Read the Question

Count the brackets and look at what is in front.

You see	You do
A term × two brackets	brackets first, then the term
A bracket squared	write it twice
“Show that … $a x^{3} + b x^{2} + c x$ ”	expand fully, then read off a, b and c
Three brackets	two at a time (Expand and Simplify)
Two brackets only	Expanding Two Brackets
Make it a product	Simple Factorising

Key idea

The answer has an x cubed term. If it does not, something was missed.

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