Ask the questions in order. Rules and names from term1.pdf pp.1–4 (“Factorising” 1–5) and pp.23–24 (cubics).
1
Is there something in EVERY term? (a number, a letter, or a whole bracket; watch for (8 − 4x) = −(4x − 8))
YES → Common factor (HCF) first. Then restart at 2 with what’s left in the bracket.
no / done ↓ now count the terms
2
TWO terms with a minus, both perfect squares? A² − B²
(squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…; even powers x², x⁴, a¹⁶; or (bracket)²)
YES → Difference of squares
A² − B² = (A − B)(A + B)
3
TWO terms, both perfect cubes? A³ ± B³
(cubes: 1, 8, 27, 64, 125, 216…; powers divisible by 3: x³, x⁶, x¹², y⁹)
YES → Sum/difference of cubes
A³ + B³ = (A + B)(A² − AB + B²)
A³ − B³ = (A − B)(A² + AB + B²)
4
THREE terms, highest power 2: ax² + bx + c
YES → Trinomial: find two numbers that multiply to a×c and add to b (cross method).
5
FOUR terms? Try pairs (2+2). If the squares stand out, pair them; if three terms form a perfect square, use 3+1.
YES → Grouping, then look for a common bracket (or a difference of squares).
6
Cubic ax³ + bx² + cx + d and steps 1 & 5 don’t work?
→ Factor theorem: try factors of d (±1, ±2…) until f(k) = 0; then synthetic division; then factorise the quadratic.
↓ always
✓
Check every bracket again (“Always check if you can factorise further”, term1 p.24). E.g. c⁴ − 81 → (c² − 9)(c² + 9) → (c − 3)(c + 3)(c² + 9).
A sum of squares A² + B² does not factorise (over real numbers).
| You see… | Think… |
|---|
| minus between two squares | Difference of squares |
| “+” between two squares | Stop: doesn’t factorise (unless there is a common factor) |
| 8, 27, 64, 125, x³, x⁶, x¹² | Cubes |
| x², x and a number | Trinomial |
| 4 terms, letters like p, q, b mixed in | Grouping |
| x³ and x², x, number with no grouping | Factor theorem + synthetic division |
| ( )(…) + ( )(…) with brackets that match up to a sign | Common (bracket) factor |