Algebraic Manipulation

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Simplify expressions by collecting like terms
  2. Expand products of algebraic expressions
  3. Factorise by extracting common factors
  4. Expand products of two brackets involving one variable and factorise fully
  5. Factorise expressions of the form: ax + bx + kay + kby; a²x² − b²y²; a² + 2ab + b²; ax² + bx + c; ax³ + bx² + cx
  6. Complete the square for expressions in the form ax² + bx + c
  7. Expand products of more than two brackets and factorise fully

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1. Simplifying Expressions by Collecting Like Terms

What does "simplify" mean?

To simplify an expression means to write it in its shortest, clearest form. You do this by combining like terms - terms that have exactly the same variable parts.

Like terms are terms that contain the same letters raised to the same powers. For example:

  • 3x and 5x are like terms (both have just x)
  • 2a² and 7a² are like terms (both have a²)
  • 3x and 3y are NOT like terms (different letters)
  • 5x and 5x² are NOT like terms (different powers)

How to simplify:

  1. Identify all like terms in the expression
  2. Add or subtract the numbers in front (called coefficients) of like terms
  3. Keep the variable part the same

Example 1: Simplify 2a + 3b + 5a − 9b

  • Like terms with a: 2a and 5a → 2a + 5a = 7a
  • Like terms with b: 3b and −9b → 3b − 9b = −6b
  • Answer: 7a − 6b

Example 2: Simplify 2a² + 3ab − 1 + 5a² − 9ab + 4

  • Like terms with a²: 2a² and 5a² → 7a²
  • Like terms with ab: 3ab and −9ab → −6ab
  • Numbers: −1 and 4 → 3
  • Answer: 7a² − 6ab + 3

Key point: When simplifying, always write your final answer with like terms collected together. Don't leave 3x + 5y + 2x as your answer - simplify it to 5x + 5y.

2. Expanding Brackets

Expanding brackets means removing the brackets by multiplying everything inside by what's outside. This is the opposite of factorising.

2.1 Expanding Single Brackets

The rule: Multiply the term outside the bracket by each term inside the bracket.

Example 1: Expand 3x(2x − 4y)

  • Multiply 3x by 2x: 3x × 2x = 6x²
  • Multiply 3x by −4y: 3x × (−4y) = −12xy
  • Answer: 6x² − 12xy

Example 2: Expand 4x(2x − 3)

  • 4x × 2x = 8x²
  • 4x × (−3) = −12x
  • Answer: 8x² − 12x

Watch out for negative signs!

When expanding −7x(4 − 5y):

  • (−7x) × 4 = −28x
  • (−7x) × (−5y) = +35xy (negative × negative = positive)
  • Answer: −28x + 35xy

Expanding and simplifying multiple brackets:

When you have more than one set of brackets being added or subtracted, expand each separately, then collect like terms.

Example: Expand and simplify 2(x + 5) + 3x(x − 8)

Step 1: Expand first bracket: 2 × x + 2 × 5 = 2x + 10

Step 2: Expand second bracket: 3x × x + 3x × (−8) = 3x² − 24x

Step 3: Combine: 2x + 10 + 3x² − 24x

Step 4: Collect like terms: 3x² − 22x + 10

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