Expanding Algebraic Expressions
Every question with a fully worked solution.
- Expanding Algebraic Expressions - Video - Expanding & Simplifying Single Brackets Watch
Theory
To expand a bracket, multiply every term inside it by the term outside. This Year 9 maths page uses the distributive law with positive and negative numbers and pronumerals outside the bracket, then shows how to collect like terms after expanding two brackets.
To expand means to remove the brackets by multiplying. The distributive law says the term outside multiplies every term inside: \(a(b + c) = ab + ac\).
The sign in front of the bracket belongs to the term outside. A negative outside changes the sign of every term inside. A bare minus sign, as in \(-(x + 2)\), means \(-1\).
Like terms have exactly the same pronumeral part, such as \(5x\) and \(-2x\). They can be added or subtracted. \(x^2\) and \(x\) are not like terms.
To simplify after expanding, collect the like terms into one term each.
The distributive law:
With a negative outside, every sign inside flips:
| Product | Sign | Example |
|---|---|---|
| positive \(\times\) positive | \(+\) | \(2 \times 5 = 10\) |
| positive \(\times\) negative | \(-\) | \(2 \times (-5) = -10\) |
| negative \(\times\) negative | \(+\) | \((-2) \times (-5) = 10\) |
How to expand and simplify
- Find the term outside each bracket, including its sign.
- Multiply it by every term inside. Multiply numbers with numbers and pronumerals with pronumerals.
- Write all the products out in one line.
- Collect like terms, usually with the highest power first.
- Check by substituting a small number, such as \(x = 1\), into the question and the answer.
(i) \(6(2k - 5)\) and
(ii) \(-(3x - 2y + 1)\).
(i) Multiply each term inside by \(6\):
| \(6 \times 2k\) | \(=\) | \(12k\) |
| \(6 \times (-5)\) | \(=\) | \(-30\) |
| \(6(2k - 5)\) | \(=\) | \(\boldsymbol{12k - 30}\) |
(ii) A bare minus sign means \(-1\). It multiplies all three terms, so every sign flips:
| \(-(3x - 2y + 1)\) | \(=\) | \(\boldsymbol{-3x + 2y - 1}\) |
Three terms inside means three multiplications.
Multiply the numbers and the pronumerals separately. \(a \times a = a^2\).
| \(-2a \times 4a\) | \(=\) | \(-8a^2\) |
| \(-2a \times (-3)\) | \(=\) | \(+6a\) |
| \(-2a(4a - 3)\) | \(=\) | \(\boldsymbol{-8a^2 + 6a}\) |
The second product is positive, because a negative times a negative is positive.
The second bracket is multiplied by \(-2\), sign included:
| \(5(x + 3)\) | \(=\) | \(5x + 15\) |
| \(-2(x - 7)\) | \(=\) | \(-2x + 14\) |
Write it all out, then collect the \(x\) terms and the numbers:
| \(\) | \(=\) | \(5x + 15 - 2x + 14\) |
| \(\) | \(=\) | \(\boldsymbol{3x + 29}\) |
Check with \(x = 1\): \(5 \times 4 - 2 \times (-6) = 32\), and \(3 + 29 = 32\).
Expand each bracket. The second one is multiplied by \(-n\), so both of its signs change:
| \(3n(n + 2)\) | \(=\) | \(3n^2 + 6n\) |
| \(-n(2n - 5)\) | \(=\) | \(-2n^2 + 5n\) |
Collect \(n^2\) terms with \(n^2\) terms and \(n\) terms with \(n\) terms:
| \(\) | \(=\) | \(3n^2 - 2n^2 + 6n + 5n\) |
| \(\) | \(=\) | \(\boldsymbol{n^2 + 11n}\) |
\(n^2\) and \(n\) are not like terms, so the answer has two terms.
Common pitfalls
Frequently asked questions
How do you expand brackets?
Multiply the term outside the bracket by every term inside it, keeping track of the signs, then collect any like terms.
What happens when you expand a bracket with a negative in front?
The negative multiplies every term inside, so every sign inside the bracket changes. For example \(-2(x - 3) = -2x + 6\).
What is the distributive law?
It says \(a(b + c) = ab + ac\). The term outside a bracket is shared out to multiply each term inside.
What are like terms?
Terms with exactly the same pronumerals and powers, such as \(5x\) and \(2x\). \(x^2\) and \(x\) are not like terms.