Algebraic manipulation — simplify, expand, factorise

This is the most heavily-tested algebra topic on WJEC. Foundation tests collect-like-terms and single-bracket expansion; Intermediate adds binomial expansion and basic factorisation; Higher tests two-bracket factorisation including non-monic quadratics.

Collecting like terms

Like terms have the same variable parts (same letters and powers).

3x + 5y − x + 2y = (3x − x) + (5y + 2y) = 2x + 7y.

Expanding a single bracket

Multiply every term inside by the term outside.

5(2x + 3) = 10x + 15. −3(x − 4) = −3x + 12. Be careful with the sign.

Expanding two brackets (binomials)

(x + a)(x + b) — use FOIL or grid:

• F: x × x = x^2
• O: x × b = bx
• I: a × x = ax
• L: a × b = ab

Total: x^2 + (a+b)x + ab.

(x + 3)(x − 5) = x^2 − 5x + 3x − 15 = x^2 − 2x − 15.

Factorising single bracket

Find the highest common factor (HCF) of all terms; pull it out.

8x + 12 → HCF = 4 → 4(2x + 3).

6x^2 − 9x → HCF = 3x → 3x(2x − 3).

Factorising quadratics: x^2 + bx + c

Find two numbers that multiply to c and add to b.

x^2 + 7x + 12: 3 × 4 = 12 and 3 + 4 = 7. So (x + 3)(x + 4).

x^2 − 5x + 6: −2 × −3 = 6 and −2 + −3 = −5. So (x − 2)(x − 3).

x^2 + 2x − 15: −3 × 5 = −15 and −3 + 5 = 2. So (x − 3)(x + 5).

Difference of two squares (Higher)

a^2 − b^2 = (a − b)(a + b).

x^2 − 49 = (x − 7)(x + 7). 4x^2 − 9 = (2x − 3)(2x + 3).

Non-monic quadratics: ax^2 + bx + c (Higher)

Multiply a × c, find two numbers that multiply to ac and add to b, split the middle term, factorise by grouping.

2x^2 + 7x + 3: ac = 6; need 6 and 1 (sum 7). Split: 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

WJEC exam tip

The marker awards M1 for the method and A1 for the answer. Showing the FOIL grid, even crudely, secures the M1 even if your final A1 is wrong.