Algebraic Expressions
What is an Algebraic Expression?
An algebraic expression contains numbers, variables (letters representing unknown values), and operations. Unlike equations, expressions do not have an equals sign.
Examples: 3x + 5, 2a² - b, 4(x + 3)
Expressions represent a quantity whose value depends on the variable. When you substitute a value for the variable, you can calculate the result.
Key Vocabulary
| Term | Meaning | Example in 5x² - 3x + 7 |
|---|---|---|
| Term | A single number, variable, or product | 5x², -3x, 7 |
| Coefficient | Number multiplying the variable | 5 in 5x², -3 in -3x |
| Variable | The unknown letter | x |
| Constant | A fixed number with no variable | 7 |
| Like terms | Same variable and same power | 3x and 5x |
Identifying Like Terms
Like terms have the same variable(s) raised to the same power. Only like terms can be combined.
- 3x and 5x are like terms ✓ (both have x to the power 1)
- 4x² and 2x² are like terms ✓ (both have x²)
- 3x and 3x² are NOT like terms ✗ (different powers)
- 3x and 5y are NOT like terms ✗ (different variables)
Simplifying Expressions
Collect like terms by adding or subtracting their coefficients:
Example: 4x + 3y + 2x - y
- Group x-terms: 4x + 2x = 6x
- Group y-terms: 3y - y = 2y
- Answer: 6x + 2y
Example: 5a - 3b + 2a + 4b - a
- a-terms: 5a + 2a - a = 6a
- b-terms: -3b + 4b = b
- Answer: 6a + b
You cannot simplify unlike terms — 3x + 5y cannot be simplified further.
Expanding Brackets
Multiply every term inside the bracket by the term outside:
Example: 3(2x + 5)
- 3 × 2x = 6x
- 3 × 5 = 15
- Answer: 6x + 15
Example: -2(x - 4)
- -2 × x = -2x
- -2 × (-4) = +8
- Answer: -2x + 8
Take special care with negative signs outside the bracket — they change the sign of every term inside.
Expanding and Simplifying
Expand all brackets first, then collect like terms.
Example: 2(x + 4) + 3(x - 1)
- Expand: 2x + 8 + 3x - 3
- Collect: (2x + 3x) + (8 - 3) = 5x + 5
Substitution
Replace each variable with its given value and calculate. Follow the order of operations (BIDMAS/BODMAS).
Example: Find 2x² - x + 4 when x = 3
- 2(3²) - 3 + 4
- 2(9) - 3 + 4
- 18 - 3 + 4 = 19
Example: Find a² + 2b when a = 2, b = -3
- (2²) + 2(-3)
- 4 + (-6) = -2
Factorising Expressions
Factorising is the reverse of expanding — find the highest common factor and write it outside a bracket.
Example: 6x + 9
- HCF of 6 and 9 is 3
- 6x + 9 = 3(2x + 3)
Example: 10x² + 15x
- HCF of 10 and 15 is 5; both terms have x
- 10x² + 15x = 5x(2x + 3)
Writing Expressions from Words
Translating word problems into algebra is a key skill:
| Word phrase | Algebraic expression |
|---|---|
| 5 more than x | x + 5 |
| 4 less than y | y - 4 |
| 3 times a number n | 3n |
| A number divided by 7 | n/7 |
| Square of a number | n² |
Common Mistakes
- Combining unlike terms: 3x + 4y ≠ 7xy. Never add terms with different variables.
- Forgetting to multiply all terms in the bracket: 3(x + 2) ≠ 3x + 2.
- Sign errors when expanding: -2(x - 3) = -2x + 6, not -2x - 6.
- Squaring only the number, not the variable: 2x² means 2(x²), not (2x)².
Tips and Tricks
- Underline like terms in the same way before collecting them (e.g., underline all x-terms, circle all y-terms).
- When substituting negative numbers, always put them in brackets: (−3)² = 9, not −3² = −9.
- To check a factorisation, expand your answer — you should get back the original expression.
- The number of terms in a simplified expression tells you something: one term = monomial, two = binomial, three = trinomial.