Expanding Expressions and Simplifying
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.