Expanding Triple Brackets (Cambridge (CIE) IGCSE Maths)

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Written by: Mark Curtis

Reviewed by: Dan Finlay

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Expanding Three Brackets

How do I expand three brackets?

Multiply out any two of the brackets using a standard method and simplify
Then multiply the resulting expression by the third (unused) bracket

This step often looks like (x + a)(x² + bx + c)
Every term in the first bracket must be multiplied with every term in the second bracket
A grid can help to keep track of all the terms
- E.g. (x + 2)(x² + 3x + 1)
  x²
  +3x
  +1
  x
  x³
  3x²
  x
  +2
  2x²
  6x
  2

	x²	+3x	+1
x	x³	3x²	x
+2	2x²	6x	2

Add all the terms inside the grid together
- x³ + 2x²+ 3x² + 6x + x + 2
Simplify by collecting any like terms
- x³ + 5x² + 7x + 2

Worked Example

Expand $(2 x - 3) (x + 4) (3 x - 1)$ .

Expand and simplify the first two brackets, for example using the FOIL method

$\begin{array}{rcl} (2 x - 3) (x + 4) \\ = & 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4 \\ = & 2 x^{2} + 8 x - 3 x - 12 \\ = & 2 x^{2} + 5 x - 12 \end{array}$

Rewrite the original expression with the first two brackets expanded

$(2 x^{2} + 5 x - 12) (3 x - 1)$

Multiply all of the terms in the first set of brackets by all of the terms in the second set of brackets

A grid can help when there are many terms to multiply together (e.g. write $2 x^{2} + 5 x - 12$ in the vertical column and $3 x - 1$ in the horizontal column, then multiply corresponding terms)

	$3 x$	$- 1$
$2 x^{2}$	$6 x^{3}$	$- 2 x^{2}$
$5 x$	$15 x^{2}$	$- 5 x$
$- 12$	$- 36 x$	$12$

Write out the multiplied terms

$6 x^{3} - 2 x^{2} + 15 x^{2} - 5 x - 36 + 12$

Collect the like terms to simplify

$6 x^{3} + 13 x^{2} - 41 x + 12$

Last updated: 17 October 2024

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