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Algebraic manipulation

Revise Algebraic manipulation for Mathematics A 4MA1 (O Level) — revision notes, 566 practice questions and instant AI marking.

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Algebraic manipulation

The big idea: Expanding and factorising are the same move in opposite directions. Every technique here, from collecting like terms to completing the square, rewrites an expression without changing its value.

Summary — What This Chapter Covers

Substitute numbers for letters, and collect like terms (same letters, same powers).
Expand a single bracket; factorise fully by taking out the highest common factor.
Expand two brackets and factorise x² + bx + c into two brackets.
Higher: expand three brackets, factorise ax² + bx + c, and simplify algebraic fractions.
Higher: complete the square, a(x + b)² + c, and write algebraic proofs.

1. Substituting and Collecting Like Terms

Foundation & Higher

Evaluate, then tidy up

To evaluate an expression, replace each letter by its value, using brackets around negatives. Like terms have exactly the same letters raised to the same powers: 3a² and −a² are like terms; 3a² and 3a are not.

Worked Example

Evaluate 5p + 2q when p = 3 and q = 4.

Practice Question 1

Simplify (a) 5x + 3y − 2x + 7y (b) 4a² + 3a − a² − 5a

2. Single Brackets and Common Factors

Foundation & Higher

Expand, and factorise fully

To expand, multiply every term inside the bracket by the term outside. Factorising reverses this: find the highest common factor (HCF) of the number parts and of the letters, and put it outside. "Factorise fully" means nothing common is left inside.

Worked Example

Expand 3x(2x + 5).

Worked Example

Factorise fully 28xy + 12y.

Practice Question 2

Expand and simplify 4(2x − 3) − 3(x − 5).

Practice Question 3

Factorise fully 15a²b − 10ab².

3. Two Brackets and Factorising Quadratics

Foundation & Higher

x² + bx + c

Expanding (x + p)(x + q) gives x² + (p + q)x + pq. So to factorise x² + bx + c, find two numbers that multiply to c and add to b. A quadratic expression is one whose highest power of x is 2.

Pattern (x + p)(x + q) = x² + (p + q)x + pq · x² − a² = (x + a)(x − a)

Worked Example

Expand and simplify (x + 8)(x − 5).

Worked Example

Factorise x² + 10x + 24.

Practice Question 4

Factorise (a) x² − 2x − 15 (b) x² − 49 (c) Expand (x − 7)²

4. Three Brackets and Harder Quadratics

Higher tier

Expand (x + a)(x + b)(x + c); factorise ax² + bx + c

For three brackets, expand two first, simplify, then multiply by the third. To factorise ax² + bx + c with a ≠ 1, find two numbers that multiply to ac and add to b, split the middle term, then factorise in pairs.

Worked Example

Expand and simplify (x + 2)(x + 3)(x − 1).

Worked Example

Factorise 6x² − 5x − 6.

5. Algebraic Fractions

Higher tier

Simplify, add and subtract

factorise top and bottom fully factors

Worked Example

Simplify (2x² + 3x)/(4x² − 9).

Worked Example

Express 3/(x + 2) + 1/(x − 1) as a single fraction in its simplest form.

Practice Question 5

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Writing a quadratic as a(x + b)2 + c
Show it is true for every value

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Algebraic manipulation

Algebraic manipulation

Evaluate, then tidy up

Expand, and factorise fully

x2 + bx + c

Expand (x + a)(x + b)(x + c); factorise ax2 + bx + c

Simplify, add and subtract

Past papers that test Algebraic manipulation

More Mathematics A topics

x² + bx + c

Expand (x + a)(x + b)(x + c); factorise ax² + bx + c