Expanding Brackets

Rules for Expanding Brackets

Expanding brackets, or “multiplying out”, involves multiplying every term inside the bracket by the number or term on the outside with the aim of removing the set of brackets.

The formula for expanding a single bracket is \[a(b+c) = ab + ac\]

The formula for expanding a double bracket is \[(a+b)(c+d) = a(c+d)+b(c+d)=ac + ad + bc + bd\]

This last formula for the product is often referred to as the FOIL method:

Multiply the First terms, Outside terms, Inside terms, Last terms.

$a$ and $c$ are the first terms, $a$ and $d$ are the outside terms, $b$ and $c$ are the inside terms and $b$ and $d$ are the last terms.

Worked Examples

Example 1

Expand $2x(xy-3x^2)$.

Solution

Start by multiplying $2x$ by the first term inside the bracket, then by the second. \[2x(xy-3x^2) = 2x \times xy-2x\times 3x^2=2x^2y - 6x^3\]

Example 2

Expand $(3x-4)(8-2x)$.

Solution

Start by multiplying $3x$ by $8$ and $-2x$, then multiply $-4$ by $8$ and $-2x$. \begin{align} (3x-4)(8-2x)&= 3x(8-2x)-4(8-2x)\\ &=24x -6x^2 -32 +8x\\ &=32x -6x^2-32 \end{align}

Video Examples

Example 1

Prof. Robin Johnson expands the expression $x(2x-1)(2-x)$.

Example 2

Prof. Robin Johson expands the expression $(x-y)(x+y)$ and talks about the difference of two squares, which comes into use when factorising.

Test Yourself

Test yourself: Numbas test on expanding brackets

Test yourself: Numbas test on Algebraic Manipulation

External Resources