Expanding brackets
Introduction
Expanding brackets involves removing the brackets from an expression by multiplying out the brackets. This is achieved by multiplying every term inside the bracket by the term outside the bracket.
When multiplying out double brackets, every term in the first pair of brackets must be multiplied by each term in the second.
When expanding brackets, be very careful when dealing with negative numbers.
Example questions on expanding and simplifying expressions
Example 1 - Expanding a single pair of brackets
a) Expand: \(3(x + 6)\).
a) Remember to multiply every term inside the brackets by the term outside:
\(3(x + 6) = 3 × x + 3 × 6 = 3x + 18\).
b) Expand: \(6(4a - 10)\).
b) Remember to multiply every term inside the brackets by the term outside:
\(6(4a - 10) = 24a - 60\).
c) Expand: \(3xy(2x + y^2)\).
c) When multiplying more complicated terms, multiply the numbers first followed by the letters:
\(3xy(2x + y^2) = 6x^{2}y + 3xy^{3}\).
Example 2 - Expanding and simplifying brackets
a) Expand and simplify \(2(3x + 4) + 4(x - 1)\).
Multiply each bracket out first, then collect the like terms: \begin{aligned} 2(3x + 4) + 4(x - 1) & = 6x + 8 + 4x - 4\\ & = 10x + 4\\ \end{aligned}
b) Expand and simplify \(7(3n - 9) - 4(6 - 4n)\).
Be very careful when multiplying out brackets with lots of negative signs: \begin{aligned} 7(3n - 9) - 4(6 - 4n) & = 21n - 63 - 24 + 16n\\ & = 37n - 87\\ \end{aligned}
Example 3 - Expanding double brackets
Expand and simplify \((a + b)(c + d)\).
When multiplying out double brackets, each terms in the first bracket must be multiplied by each term in the second:
\((a + b)(c + d) = ac + ad + bc + bd\).
Example 4 - Expanding and simplifying quadratic expressions
a) Expand and simplify \((x + 4)(x + 3)\).
When multiplying \(x\) by another \(x\) you will end up with an \(x^2\) term: \begin{aligned} (x + 4)(x + 3) & = x^2 + 3x + 4x + 12\\ & = x^2 + 7x + 12\\ \end{aligned}
b) Expand and simplify \((3x - 10)(5x - 9)\).
Remember, when multiplying two negative terms you will get a positive: \begin{aligned} (3x - 10)(5x - 9) & = 15x^2 - 27x - 50x + 90\\ & = 15x^2 - 77x + 90\\ \end{aligned}
Worksheets to practise expanding and simplifying expressions
Try these worksheets to practise your skills.