8.3 Expanding brackets — Learn

Prerequisites: collecting like terms and multiplying positive and negative numbers.

By the end of this section you can: expand a single bracket, including when the factor is negative, then collect like terms.

Covers: A4.

Learn the method

Expanding means multiplying each term inside brackets by the factor outside. This uses the distributive law.

For $3(x+4)$, multiply $3$ by both terms: $3\times x+3\times4=3x+12$.

A negative factor multiplies every term. For $-2(y-5)$, the products are $-2y$ and $+10$.

If terms appear outside the bracket, expand first, then collect like terms.

Worked example. Expand $-3(2x-5)$.

$-3(2x-5)$

$=(-3)(2x)+(-3)(-5)$ — multiply $-3$ by both terms.

$=-6x+15$

Watch out. A common error is to multiply only the first term in a bracket. Multiply every term, then expand to check.

Guided investigation: frame area

A rectangular frame has outer dimensions $x+6$ and $x+4$. Its inner rectangle has dimensions $x$ and $x+2$. The shaded frame area is outer area minus inner area.

Predict: before using the figure, predict the simplified expression for the frame area.

Test: use the controls for $x$ and compare the two area calculations.

Explain: write why the $x^2$ terms cancel, and check your predicted expression at $x=2$.

You should now be able to

• expand a single bracket, including when the factor is negative, then collect like terms.