Expanding and Simplifying
Expanding uses the distributive law to remove brackets: every term inside is multiplied by the term outside. This NSW Year 9 Advanced maths subtopic expands and then collects like terms (NESA outcome MA5‑ALG‑C‑01), with careful attention to negative signs.
You will expand expressions such as \(-3(5x^2 + 2y + 4)\) and \(7 - 4(2x - 1)\), then expand two brackets and simplify, as in \(3r(r + 4) - r(6r - 1)\). Harder questions produce cubic terms, write the planting area of a garden bed in expanded form, find an unknown constant from a simplified answer, and show that one expression equals zero for every value of its pronumerals. It is part of Year 9 Advanced Algebra: Expanding and Factorising.
Every question with a fully worked solution.
- Expanding and Simplifying - Video - Expanding & Simplifying Single Brackets Watch
Theory
To expand a bracket, multiply every term inside by the term outside; then simplify by collecting like terms. This Year 9 Advanced maths page covers negative and pronumeral multipliers, two or more brackets, area expressions and finding an unknown coefficient.
To expand means to remove the grouping symbols by multiplying. The distributive law says the term outside multiplies every term inside: \(a(b + c) = ab + ac\).
The sign in front of a term belongs to it. In \(6 - 5(m - 2)\) the multiplier is \(-5\), so \(-5 \times (-2) = +10\). A lone minus sign in front of a bracket means \(-1\): \(-(y - 8) = -y + 8\).
To simplify, collect like terms: terms with exactly the same pronumerals and powers. \(4x^2\) and \(-x^2\) are like terms; \(4x^2\) and \(4x\) are not.
When the term outside contains a pronumeral, use the index laws: \(2y^2 \times 5y = 10y^3\) and \(-3p \times 4q = -12pq\).
The distributive law, with the signs:
| Signs | Product | Example |
|---|---|---|
| \(+ \times +\) | \(+\) | \(4 \times 3y = 12y\) |
| \(+ \times -\) | \(-\) | \(4 \times (-3y) = -12y\) |
| \(- \times -\) | \(+\) | \(-4 \times (-3y) = 12y\) |
How to expand and simplify
- Mark the multiplier for each bracket, including its sign.
- Multiply it by every term inside. Count: three terms inside means three products.
- Collect like terms, highest power first.
- Check by substituting a small value such as \(x = 2\) into the question and the answer.
Problems built on expanding
- Area: write each length as an expression, multiply, and subtract any piece cut out. Substitute to find a numerical area.
- Unknown coefficient: expand and collect, then match the coefficients with the given answer and solve.
- Always the same value: if every term cancels when you expand, the expression has the same value for all values of the pronumerals. For example \(x(y + 2) - y(x - 3) - 2x - 3y\) \(= 0\).
The \(-2\) multiplies both terms in the bracket. Then collect like terms.
| \(5k - 2(3k - 7) + 4\) | \(=\) | \(5k - 6k + 14 + 4\) |
| \(\) | \(=\) | \(\boldsymbol{18 - k}\) |
\(-2 \times (-7) = +14\). Check with \(k = 1\): \(5 - 2(-4) + 4 = 17\) and \(18 - 1 = 17\).
Expand each bracket separately, keeping the sign in front of each multiplier.
| \(3a(2a - b + 4)\) | \(=\) | \(6a^2 - 3ab + 12a\) |
| \(-2a(a + 5)\) | \(=\) | \(-2a^2 - 10a\) |
| \(\text{Total}\) | \(=\) | \(6a^2 - 2a^2 - 3ab + 12a - 10a\) |
| \(\) | \(=\) | \(\boldsymbol{4a^2 - 3ab + 2a}\) |
\(3a \times 2a = 6a^2\): multiply the numbers and add the indices. \(a^2\), \(ab\) and \(a\) are all different kinds of term.
(i) Write the area of grass left, in expanded form.
(ii) Find this area when \(x = 3\).
Grass area \(=\) whole rectangle \(-\) pond.
(i)
| \(A\) | \(=\) | \(2x(4x + 3) - x(x + 2)\) |
| \(\) | \(=\) | \(8x^2 + 6x - x^2 - 2x\) |
| \(\) | \(=\) | \(\boldsymbol{(7x^2 + 4x)}\text{ m}^2\) |
(ii)
| \(A\) | \(=\) | \(7(3)^2 + 4(3)\) |
| \(\) | \(=\) | \(63 + 12 = \mathbf{75}\text{ m}^2\) |
Check with the real sizes: the lawn is \(15 \times 6 = 90\) m\(^2\), the pond is \(5 \times 3 = 15\) m\(^2\), and \(90 - 15 = 75\).
Expand and collect, then match the \(x\)-terms and the numbers with \(4x + 28\).
| \(a(x + 3) - 2(x - 5)\) | \(=\) | \(ax + 3a - 2x + 10\) |
| \(\) | \(=\) | \((a - 2)x + (3a + 10)\) |
| \(a - 2\) | \(=\) | \(4\) |
| \(a\) | \(=\) | \(\mathbf{6}\) |
Check the numbers too: \(3 \times 6 + 10 = 28\). Both parts match, so \(a = 6\).
Common pitfalls
Frequently asked questions
How do you expand brackets?
Multiply the term outside the bracket by every term inside it. For example \(3(2x + 5) = 6x + 15\). Then collect any like terms.
What happens when you expand with a negative number?
The negative multiplies every term, so every sign inside changes. For example \(-4(x - 2) = -4x + 8\), because negative times negative is positive.
What does a minus sign in front of a bracket mean?
It means multiply by \(-1\). So \(-(3x - 7) = -3x + 7\). Every term inside changes sign.
How do you check that you expanded correctly?
Substitute a small number, such as x = 2, into the original expression and into your answer. If the two results are different, there is an error.
What are like terms?
Like terms have exactly the same pronumerals raised to the same powers, such as 5ab and -2ab. Only like terms can be added or subtracted.