Expanding Expressions and Simplifying
Expanding removes brackets using the distributive law, \(a(b + c) = ab + ac\). In NSW Year 8 maths every term inside the bracket is multiplied by the term outside, sign and all, and like terms are then collected (NESA outcome MA4‑ALG‑C‑01).
You will expand brackets with negative and pronumeral terms in front, such as \(-x(3x + 2)\) and \(5m(3m - 4)\), take care when a bracket is subtracted, as in \(24 - 7(x + 3)\), and simplify sums of two expanded brackets. You will check an answer by substituting a value. The total cost of a canteen order and the combined area of two rectangles are both found by expanding. It is part of Year 8 Algebra.
Theory
The distributive law: \(a(b + c) = a \times b + a \times c = ab + ac\). To expand, multiply every term inside the bracket by the term outside, then collect like terms: \(3 + 2(x + 5) = 3 + 2x + 10 = 2x + 13\). This Year 8 maths page shows how.
The distributive law. \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\). Removing the brackets this way is called expanding.
The outside term keeps its sign. A negative outside multiplies every term: \(-3(x + 4) = -3x - 12\) and \(-5(p - 2) = -5p + 10\), because a negative times a negative is positive.
A term outside. Multiply the numbers, then the pronumerals: \(2x(x + 5) = 2x^2 + 10x\).
Subtracting a bracket. The minus belongs to the term in front of the bracket: \(9 - 2(x - 3) = 9 - 2x + 6 = 15 - 2x\). A bare minus means \(-1\): \(-(x - 4) = -x + 4\), and \(-x(4x + 3) = -4x^2 - 3x\).
Then collect like terms, squared term first, then the pronumeral term, then the number. Check by substituting the same number into the question and the answer.
The distributive law:
How to expand and simplify
- Multiply every term inside each bracket by the term in front (with its sign).
- Collect the like terms.
- Check by substituting a number.
| \(3 + 2(x + 5)\) | \(=\) | \(3 + 2x + 2 \times 5\) |
| \(\) | \(=\) | \(3 + 2x + 10\) |
| \(\) | \(=\) | \(2x + 13\) |
\(2x + 13\)
| \(\) | \(=\) | \(2x - 8 - x^2 - 2x\) |
| \(\) | \(=\) | \(-x^2 - 8\) |
\(-x^2 - 8\) (the \(2x\) terms cancel)
(i) \(4(3p - 5)\)
(ii) \(-2(a + 7b)\)
| \(4(3p - 5)\) | \(=\) | \(12p - 20\) |
| \(-2(a + 7b)\) | \(=\) | \(-2a - 14b\) |
(i) \(12p - 20\)
(ii) \(-2a - 14b\)
| \(3k(k - 2)\) | \(=\) | \(3k^2 - 6k\) |
| \(3k^2 - 6k + 5k\) | \(=\) | \(3k^2 - k\) |
\(3k^2 - k\). Check with \(k = 4\): \(12 \times 2 + 20 = 44\) and \(48 - 4 = 44\).
| \(P\) | \(=\) | \(2(3x - 1) + 2(x + 4)\) |
| \(\) | \(=\) | \(6x - 2 + 2x + 8\) |
| \(\) | \(=\) | \(8x + 6\) |
\((8x + 6)\) cm. Check with \(x = 3\): sides \(8\) and \(7\), so \(P = 30\), and \(8 \times 3 + 6 = 30\).
| \(4c(c + 3)\) | \(=\) | \(4c^2 + 12c\) |
| \(-2c(c - 5)\) | \(=\) | \(-2c^2 + 10c\) |
| \(\) | \(=\) | \(2c^2 + 22c\) |
\(2c^2 + 22c\). Check with \(c = 3\): \(12 \times 6 - 6 \times (-2) = 84\) and \(18 + 66 = 84\).
Common pitfalls
Frequently asked questions
What is the distributive law?
\(a(b + c) = ab + ac\): the term outside multiplies every term inside.
How do you expand a negative times a bracket?
Multiply each term by the negative: \(-2(x - 3) = -2x + 6\).
How do you expand x(x + 4)?
\(x \times x + x \times 4 = x^2 + 4x\).
What do you do after expanding?
Collect like terms, writing the answer in descending powers.
How can I check an expansion?
Substitute a number into the original and your answer; both must give the same value.