A4: Simplifying Expressions
Simplify and manipulate algebraic expressions: collecting like terms, expanding brackets, factorising
📋 Key Concepts
Simplifying: Writing expressions in their simplest form by combining like terms and removing unnecessary brackets.
Key Skills:
- Collecting like terms
- Expanding brackets
- Factorising expressions
📝 Collecting Like Terms
Like terms: Terms with the same variable(s) and same powers. Only like terms can be combined.
Example 1
Which are like terms?
a) 3x and 5x → Yes (both have x)
b) 2x² and 3x → No (different powers)
c) 4ab and 7ab → Yes (both have ab)
d) 2xy and 3yx → Yes (same variables, order doesn't matter)
Example 2
Simplify: 3x + 2x + 5x
Solution: 3x + 2x + 5x = 10x
Add the coefficients: 3 + 2 + 5 = 10
Example 3
Simplify: 4a + 3b - 2a + 5b
Solution:
Collect a terms: 4a - 2a = 2a
Collect b terms: 3b + 5b = 8b
Answer: 2a + 8b
Example 4
Simplify: 5x² + 2x - 3x² + 7
Solution:
x² terms: 5x² - 3x² = 2x²
x terms: 2x
Constants: 7
Answer: 2x² + 2x + 7
📝 Expanding Single Brackets
Rule: Multiply the term outside the bracket by each term inside the bracket.
a(b + c) = ab + ac
a(b - c) = ab - ac
a(b - c) = ab - ac
Example 5
Expand: 3(x + 4)
Solution: 3(x + 4) = 3 × x + 3 × 4 = 3x + 12
Example 6
Expand: 2(5a - 3)
Solution: 2(5a - 3) = 2 × 5a - 2 × 3 = 10a - 6
Example 7
Expand: -2(x + 5)
Solution: -2(x + 5) = -2x - 10
Be careful with negative signs!
📝 Expanding Double Brackets
FOIL method: First, Outer, Inner, Last
Example 8
Expand: (x + 3)(x + 2)
Solution using FOIL:
First: x × x = x²
Outer: x × 2 = 2x
Inner: 3 × x = 3x
Last: 3 × 2 = 6
Add together: x² + 2x + 3x + 6 = x² + 5x + 6
Example 9
Expand: (2x - 1)(x + 4)
Solution:
First: 2x × x = 2x²
Outer: 2x × 4 = 8x
Inner: -1 × x = -x
Last: -1 × 4 = -4
Answer: 2x² + 8x - x - 4 = 2x² + 7x - 4
📝 Factorising
Factorising: The reverse of expanding. Find the highest common factor (HCF) and put it outside brackets.
Example 10
Factorise: 6x + 9
Solution:
HCF of 6 and 9 is 3
6x ÷ 3 = 2x
9 ÷ 3 = 3
Answer: 3(2x + 3)
Example 11
Factorise: 4x² - 8x
Solution:
HCF of 4x² and 8x is 4x
4x² ÷ 4x = x
-8x ÷ 4x = -2
Answer: 4x(x - 2)
Example 12
Factorise: 3x² + 12x
Solution:
HCF of 3x² and 12x is 3x
Answer: 3x(x + 4)
🧮 Worked Examples — Exam Style
Simplifying expressions is the grammar of algebra — every later topic assumes it. Two skills: collecting like terms (only identical letter parts combine) and expanding brackets (everything inside is multiplied).
Expand and simplify: 3(2x + 5) − 2(x − 4)
Expand each bracket: 6x + 15 − 2x + 8
Careful with the second: −2 × (−4) = +8
Collect: (6x − 2x) + (15 + 8) = 4x + 23
M1 for both expansions, A1 for 4x + 23. The minus-before-bracket case is the single most examined slip.
Factorise fully: 6x² + 9x
Highest common factor of 6x² and 9x is 3x.
6x² ÷ 3x = 2x and 9x ÷ 3x = 3
Answer: 3x(2x + 3)
M1 for 3x, A1 for 3x(2x + 3). 'Fully' means take out ALL common factors — 3(2x² + 3x) loses the mark.
Exam wording to expect: "Simplify", "Expand and simplify", "Factorise fully".
❓ Practice Questions
Q1: Simplify: 5a + 3a - 2a
Q2: Simplify: 4x + 3y - 2x + 5y
Q3: Expand: 4(x - 3)
Q4: Expand and simplify: (x + 2)(x + 5)
Q5: Factorise: 8x + 12
Q6: Factorise: 5x² - 15x
Q7: Expand and simplify: 3(2x − 1) − 2(x + 4)
Q8: Factorise fully: 10x² − 5x
✅ Answers
- 6a
- 2x + 8y
- 4x - 12
- x² + 7x + 10
- 4(2x + 3)
- 5x(x - 3)
🧮 Worked Examples — Harder Exam Style
Expand and simplify: (2x + 3)(x − 4)
Multiply every term in the first bracket by every term in the second:
2x × x = 2x² · 2x × (−4) = −8x · 3 × x = 3x · 3 × (−4) = −12
Combine: 2x² − 8x + 3x − 12 = 2x² − 5x − 12
M1 for three correct products, A1 for 2x² − 5x − 12. Check with x = 1: LHS (5)(−3) = −15; RHS 2 − 5 − 12 = −15 ✓
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. All three terms are a-terms: 5 + 3 − 2 = 6 → 6a
Q2. x-terms: 4 − 2 = 2x; y-terms: 3 + 5 = 8y → 2x + 8y. Unlike terms never combine — 2x + 8y is finished.
Q3. 4 × x and 4 × (−3): 4x − 12
Q4. FOIL: x² + 5x + 2x + 10 = x² + 7x + 10
Q5. HCF of 8 and 12 is 4 → 4(2x + 3)
Q6. HCF is 5x: 5x² ÷ 5x = x, 15x ÷ 5x = 3 → 5x(x − 3)
Q7. 6x − 3 − 2x − 8 = 4x − 11 (the −2 multiplies BOTH terms: −2 × +4 = −8)
Q8. HCF 5x → 5x(2x − 1)
📌 Key Facts to Memorise
- Like terms: same letter AND same power — 3a² and 3a are NOT like terms
- Expand: every term inside the bracket is multiplied: a(b + c) = ab + ac
- Minus before a bracket flips EVERY sign inside: −(x − 5) = −x + 5
- Factorise = expand in reverse: take out the HCF of coefficients and letters
- Double brackets: (x + a)(x + b) = x² + (a + b)x + ab
🗺️ In the Exam
Simplifying sits inside almost every algebra question as a first step: solving equations, expanding for quadratics, substitution. Standalone questions are 1–3 marks on Paper 1 (non-calculator).
Links to: solving equations (A17), quadratic graphs (A11), factorising quadratics (A18).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Only combine like terms (same variable and power)
- x² and x are NOT like terms
- When expanding, multiply every term inside by every term outside
- Always simplify your answer by collecting like terms
- Check factorising by expanding back
- Watch negative signs carefully when expanding
🧠 Problem-Solving Strategies
Problem-Solving
When simplifying, organise your work: collect like terms systematically, handle signs carefully when expanding, and always check factorising by expanding back. For double brackets, use FOIL or a grid method to ensure no terms are missed.
Multi-Step Problem
A garden has length (x + 3) metres and width (x + 1) metres. A path of width 1 metre surrounds it. Find an expression for the area of the path.
Solution:
Step 1: Garden area = (x + 3)(x + 1) = x² + 4x + 3
Step 2: Total area including path = (x + 5)(x + 3) = x² + 8x + 15
Step 3: Path area = Total - Garden = (x² + 8x + 15) - (x² + 4x + 3) = 4x + 12 m²
⚠️ Common Errors
Watch Out!
1. Wrong: 3x + 5x = 8x² Correct: 3x + 5x = 8x (add coefficients, don't change the variable part)
2. Wrong: -2(x - 3) = -2x - 6 Correct: -2(x - 3) = -2x + 6 (negative × negative = positive)
3. Wrong: Factorising 6x + 9 as 3(2x + 3) then writing 6x + 9 = 3(2x + 9) Correct: 6x + 9 = 3(2x + 3) — check: 3 × 2x = 6x, 3 × 3 = 9 ✓
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Expand and simplify (2x + 3)(x - 5). (b) Hence or otherwise, expand and simplify (2x + 3)(x - 5)(x + 1). (c) Factorise 4x² - 12x fully.
(a) (2x + 3)(x - 5) = 2x² - 10x + 3x - 15 = 2x² - 7x - 15
(b) (2x² - 7x - 15)(x + 1) = 2x³ + 2x² - 7x² - 7x - 15x - 15 = 2x³ - 5x² - 22x - 15
(c) 4x² - 12x = 4x(x - 3)
Mark scheme: (a) 2 marks for correct expansion and simplification. (b) 2 marks for multiplying correctly. (c) 2 marks for fully factorised (both 4 and x outside).
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The area of a rectangle is (x + 4)(2x - 1) cm².
(a) Expand to find the area expression.
(b) If x = 5, what is the area?
(c) Explain why x must be greater than 0.5 for the expression to make sense.
Answers: (a) 2x² - x + 8x - 4 = 2x² + 7x - 4 cm². (b) 2(25) + 7(5) - 4 = 50 + 35 - 4 = 81 cm². (c) If x ≤ 0.5, then (2x - 1) ≤ 0, giving a negative width which is impossible.
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🎬 Video Resources
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