Multiplying Terms
Every question with a fully worked solution.
- Multiplying Terms - Video - Multiplying Terms - Example 1 Watch
Theory
To multiply terms, multiply the numbers first, then the pronumerals. A pronumeral multiplied by itself is written as a power: \(2x \times 3x = 2 \times 3 \times x \times x = 6x^2\). This Year 8 maths page shows how, with negative signs and several factors.
Multiplying can be done in any order (commutative) and any grouping (associative), so a product can be rearranged into numbers together and pronumerals together.
Three steps. Multiply the coefficients, watching the signs. Write the pronumerals in alphabetical order. A pronumeral that appears more than once becomes a power: \(d \times d = d^2\), \(d^2 \times d = d^3\).
A term with no number in front has coefficient \(1\), so \(-y\) means \(-1 \times y\).
Signs. Count the negative factors: an even number gives a positive answer, an odd number a negative answer.
Writing \(\times\) between every factor makes counting easy: \(2ab \times 5bc = 2 \times a \times b \times 5 \times b \times c = 10ab^2c\).
Area and volume. Area of a rectangle \(=\) length \(\times\) width; volume of a box \(=\) length \(\times\) width \(\times\) height. A crate \(3a\) m by \(2b\) m by \(4a\) m has volume \(24a^2b\) m\(^3\).
Numbers first, then pronumerals:
| Product | Numbers | Answer |
|---|---|---|
| \(5a \times 4b\) | \(20\) | \(20ab\) |
| \(5a \times 4a\) | \(20\) | \(20a^2\) |
| \(-5a \times 4ab\) | \(-20\) | \(-20a^2b\) |
| \(-a \times (-4b)\) | \(4\) | \(4ab\) |
How to multiply terms
- Multiply the coefficients and decide the sign.
- Write the pronumerals in alphabetical order.
- Count each repeated pronumeral and write it as a power.
| \(4a \times 2b \times c\) | \(=\) | \(4 \times 2 \times a \times b \times c\) |
| \(\) | \(=\) | \(8abc\) |
\(8abc\)
\(6p \times 3q \times 2p = 6 \times 3 \times 2 \times p \times p \times q\)
| \(6 \times 3 \times 2\) | \(=\) | \(36\) |
| \(p \times p \times q\) | \(=\) | \(p^2q\) |
\(36p^2q\)
(i) \(5u \times 4v\)
(ii) \(6e \times 4e\)
| \(5u \times 4v\) | \(=\) | \(20uv\) |
| \(6e \times 4e\) | \(=\) | \(24e^2\) |
(i) \(20uv\)
(ii) \(24e^2\). Check with \(e = 3\): \(18 \times 12 = 216\) and \(24 \times 9 = 216\).
(i) \(-4h \times 6k\)
(ii) \((-5w) \times (-3w)\)
| \(-4h \times 6k\) | \(=\) | \(-24hk\) |
| \((-5w) \times (-3w)\) | \(=\) | \(15w^2\) |
(i) One negative: \(-24hk\)
(ii) Two negatives: \(15w^2\). Check with \(w = 2\): \(-10 \times (-6) = 60 = 15 \times 4\).
| \(2 \times 5 \times (-3)\) | \(=\) | \(-30\) |
| \(r \times s \times t \times t\) | \(=\) | \(rst^2\) |
\(-30rst^2\). Check with \(r = 2\), \(s = 3\), \(t = 4\): \(24 \times 10 \times (-12) = -2880\) and \(-30 \times 2 \times 3 \times 16 = -2880\).
| \(5x \times 2xy\) | \(=\) | \(10x^2y\) |
| \(2x \times xy\) | \(=\) | \(2x^2y\) |
| \(10x^2y - 2x^2y\) | \(=\) | \(8x^2y\) |
\(8x^2y\) m\(^2\). Check with \(x = 2\), \(y = 3\): \(120 - 24 = 96\) and \(8 \times 4 \times 3 = 96\).
Common pitfalls
Frequently asked questions
How do you multiply algebraic terms?
Multiply the numbers, then the pronumerals: \(3a \times 4b = 12ab\).
What is x times x?
\(x^2\), because \(x\) is used as a factor twice.
What is a negative term times a negative term?
Positive: \(-2m \times (-5n) = 10mn\).
Why write pronumerals in alphabetical order?
It makes like terms easy to spot: \(ba\) and \(ab\) are both written \(ab\).
How do you find the area of a rectangle with algebraic sides?
Multiply length by width: \(3m \times 2n = 6mn\).