Simplifying Algebraic Expressions
Every question with a fully worked solution.
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Theory
To simplify an algebraic expression, write it with as few terms as possible. This Year 9 maths page shows how to collect like terms, and how to multiply and divide terms, including negative coefficients and powers.
A term is a number, a pronumeral, or numbers and pronumerals multiplied together, such as \(-4x^2y\). The number in front is the coefficient. The sign in front of a term belongs to it.
Like terms have exactly the same pronumeral part: \(3ab\) and \(-5ba\) are like terms, but \(x^2\) and \(x\) are not. Only like terms can be added or subtracted. To simplify, add or subtract their coefficients.
A term with no number in front has a coefficient of \(1\): \(x\) means \(1x\) and \(-y\) means \(-1y\).
Any terms can be multiplied or divided, like or not. Multiply or divide the numbers, then the pronumerals: \(x^2 \times x = x^3\) and \(x^3 \div x = x^2\).
Adding like terms adds their coefficients:
Multiplying and dividing powers of the same pronumeral:
| Signs | Product or quotient | Example |
|---|---|---|
| same signs | positive | \((-2a) \times (-5b) = 10ab\) |
| different signs | negative | \(6p \times (-q) = -6pq\) |
| three negatives | negative | \((-1) \times (-1) \times (-1) = -1\) |
How to add and subtract terms
- Find the like terms. Each term keeps the sign in front of it.
- Group them together, such as \(x\) terms with \(x\) terms and numbers with numbers.
- Add or subtract the coefficients. The pronumeral part does not change.
- Write the answer, usually with the highest power first and the number last.
How to multiply and divide terms
- Signs: count the negative signs. An even number gives a positive answer, an odd number gives a negative answer.
- Numbers: multiply them, or for a division divide the top and bottom by their highest common factor.
- Pronumerals: multiply them in alphabetical order, or cancel a pronumeral that is on the top and the bottom.
- Check by substituting small numbers, such as \(a = 2\) and \(b = 3\).
(i) \(7m - 3n - 2m + 5n\)
(ii) \(3ab - 5a + 2ba + a\).
Group the like terms. Each term keeps the sign in front of it.
(i)
| \(7m - 2m\) | \(=\) | \(5m\) |
| \(-3n + 5n\) | \(=\) | \(2n\) |
| \(\therefore\) | \(\) | \(\boldsymbol{5m + 2n}\) |
(ii) \(ba\) is the same as \(ab\), so \(3ab\) and \(2ba\) are like terms. \(a\) means \(1a\).
| \(3ab + 2ab\) | \(=\) | \(5ab\) |
| \(-5a + a\) | \(=\) | \(-4a\) |
| \(\therefore\) | \(\) | \(\boldsymbol{5ab - 4a}\) |
(i) \(-5p \times 4q\)
(ii) \(2a^2b \times (-3ab) \times (-b)\).
(i) One negative sign, so the answer is negative.
| \(-5p \times 4q\) | \(=\) | \((-5 \times 4) \times pq\) |
| \(\) | \(=\) | \(\boldsymbol{-20pq}\) |
(ii) \(-b\) means \(-1 \times b\). Two negative signs, so the answer is positive. Count the \(a\)s and the \(b\)s.
| \(2 \times (-3) \times (-1)\) | \(=\) | \(6\) |
| \(a^2 \times a\) | \(=\) | \(a^3\) |
| \(b \times b \times b\) | \(=\) | \(b^3\) |
| \(\therefore\) | \(\) | \(\boldsymbol{6a^3b^3}\) |
(i) \(20m^3n \div 5mn\)
(ii) \(-12a^2b \div 18ab^3\).
Write each division as a fraction. Divide the numbers, then cancel pronumerals that are on the top and the bottom.
(i)
| \(\dfrac{20m^3n}{5mn}\) | \(=\) | \(\dfrac{20}{5} \times \dfrac{m^3}{m} \times \dfrac{n}{n}\) |
| \(\) | \(=\) | \(4 \times m^2 \times 1\) |
| \(\) | \(=\) | \(\boldsymbol{4m^2}\) |
(ii) The HCF of \(12\) and \(18\) is \(6\), and the answer is negative because the signs are different.
| \(\dfrac{-12a^2b}{18ab^3}\) | \(=\) | \(-\dfrac{12}{18} \times \dfrac{a^2}{a} \times \dfrac{b}{b^3}\) |
| \(\) | \(=\) | \(-\dfrac{2}{3} \times a \times \dfrac{1}{b^2}\) |
| \(\) | \(=\) | \(\boldsymbol{-\dfrac{2a}{3b^2}}\) |
Two \(b\)s are left on the bottom, so the answer has \(b^2\) underneath.
Do the multiplying and dividing first. Then the two terms are like terms.
| \(4a \times 3b\) | \(=\) | \(12ab\) |
| \(10ab \div 2\) | \(=\) | \(5ab\) |
| \(12ab - 5ab\) | \(=\) | \(\boldsymbol{7ab}\) |
Check with \(a = 1\) and \(b = 2\): \(4 \times 6 - 20 \div 2 = 24 - 10 = 14\), and \(7 \times 1 \times 2 = 14\).
Common pitfalls
Frequently asked questions
What are like terms in algebra?
Like terms have exactly the same pronumerals with the same powers, such as \(4xy\) and \(-2yx\). Only like terms can be added or subtracted.
How do you multiply algebraic terms?
Multiply the numbers first, then the pronumerals. A pronumeral times itself is a power, so \(a \times a = a^2\). For example \(3a \times 4ab = 12a^2b\).
How do you divide algebraic terms?
Write the division as a fraction. Divide the numbers by their highest common factor, then cancel any pronumeral that is on both the top and the bottom.
Are ab and ba like terms?
Yes. The order of multiplying does not matter, so \(ba\) is the same as \(ab\). Write the pronumerals in alphabetical order.
Why can you not add 3x and 2y?
They are not like terms, because the pronumerals are different. \(3x + 2y\) is already as simple as it can be.