6.1 Simplifying & Expanding Expressions
Key Takeaways
- A term is a number, a variable, or numbers and variables multiplied together; only like terms (same variable parts) can be collected
- Algebraic shorthand drops the multiplication sign: 3a means 3 x a, ab means a x b, and a^2 means a x a
- The distributive law says a(b + c) = ab + ac, so every term inside the brackets must be multiplied by the factor outside
- When expanding a bracket preceded by a negative sign, such as -(2x - 5), every sign inside the bracket flips
- Always check an expansion by substituting a simple number into both the original and the simplified expression
Algebra is where ICAS Mathematics stops being only about numbers and starts being about patterns and general rules. From Paper D (Year 6) number sentences with letters appear regularly, and by Papers E-F (Years 7-8) simplifying and expanding expressions is core business. Papers G-H (Years 9-10) go further, asking you to factorise and to recognise equivalent expressions. Because personal calculators are banned, all of this work is done by hand, so clean, quick written technique matters.
Variables, Terms and Expressions
A variable is a letter that stands for a number we do not know yet (or a number that can change). A term is a number, a variable, or numbers and variables multiplied together. An expression is terms joined by + or - signs.
In the expression 4x + 3y - 7 there are three terms: 4x, 3y and -7. The number in front of a variable is called the coefficient: in 4x the coefficient is 4. A term with no variable, like -7, is a constant.
Algebraic shorthand
| Notation | Meaning | Why |
|---|---|---|
| 3a | 3 x a | the x sign is dropped to avoid confusion with the letter x |
| ab | a x b | variables written together are multiplied |
| a^2 | a x a | a multiplied by itself |
| a/3 or (1/3)a | a divided by 3 | division is usually written as a fraction |
| 2(a + b) | 2 x (a + b) | a number next to brackets multiplies everything inside |
Two traps follow from this shorthand. First, 3a does NOT mean '3 and a' or 3 + a. Second, a^2 is not 2a: if a = 5, then a^2 = 25 but 2a = 10.
Collecting Like Terms
Like terms have exactly the same variable parts. 5x and -2x are like terms; 5x and 5x^2 are NOT, because the powers differ; 5x and 5y are NOT, because the variables differ. Only like terms can be added or subtracted.
To simplify, group the like terms and combine their coefficients:
- Simplify 7a + 3b - 2a + b. Group: (7a - 2a) + (3b + b) = 5a + 4b. You cannot go further, because a and b are unlike terms.
- Simplify 4x^2 + 2x - x^2 + 6x - 3. Group: (4x^2 - x^2) + (2x + 6x) - 3 = 3x^2 + 8x - 3.
Mental check: count the term types first. If the original expression has x^2 terms, x terms and constants, your answer should generally have the same three types.
Expanding Single Brackets: the Distributive Law
The distributive law says a(b + c) = ab + ac. Every term inside the brackets gets multiplied by the factor outside.
Worked example: expand 3(2x + 5).
- 3 x 2x = 6x
- 3 x 5 = 15
- Answer: 6x + 15
Worked example: expand and simplify 4(3y - 2) - 2(y - 6).
- 4(3y - 2) = 12y - 8
- 2(y - 6) = 2y - 12, and the sign before it is minus, so we subtract: -2y + 12
- Collect: 12y - 8 - 2y + 12 = 10y + 4
The sign trap ICAS loves
When a minus sign sits in front of brackets, it flips EVERY sign inside: -(2x - 5) = -2x + 5. The most common wrong answer in ICAS-style items comes from flipping only the first term and writing -2x - 5. A quick safety habit is to rewrite subtraction of a bracket as 'add the opposite' before expanding.
Expanding Two Brackets (Papers G-H)
For senior papers you may need (x + 3)(x + 5). Multiply each term in the first bracket by each term in the second (often called FOIL: First, Outside, Inside, Last):
- First: x x x = x^2
- Outside: x x 5 = 5x
- Inside: 3 x x = 3x
- Last: 3 x 5 = 15
- Collect the middle terms: x^2 + 8x + 15
Substitution Warm-up and Checking
Substitution replaces each variable with a given number. If x = 4, then 3x + 2 = 3(4) + 2 = 14. With negatives, use brackets: if a = -2, then 5 - 3a = 5 - 3(-2) = 5 + 6 = 11, not 5 - 6.
This gives you a powerful check for any simplification: pick a small number (avoid 0 and 1, which can hide errors) and substitute it into both versions. If 4(3y - 2) - 2(y - 6) and 10y + 4 both equal 24 when y = 2, your expansion is almost certainly right. ICAS 'which expression is equivalent?' questions can be solved this way even when you are unsure of the algebra: test each option with the same number.
How ICAS Phrases These Questions
Expect stems such as 'Simplify 5p - 2q + 3p + q' (answer: 8p - q, with the single -q easy to miss), 'Expand 4(2m - 3)' and 'Which expression is equivalent to 6x - 8?' where distractors include 2(3x - 4), 3(2x - 3) and 6(x - 2). The three errors examiners bank on are: combining unlike terms (writing 5x + 3 = 8x), forgetting the second multiplication inside brackets (3(2x + 5) = 6x + 5), and not flipping both signs after a minus. Read the final sign of every term before you commit to an option, and if time allows, substitute x = 2 into the original and your chosen option as a final check.
Simplify fully: 5x + 3 - 2x + 7
Expand and simplify: 2(3a - 4) - (a - 6)