Cross-method for ax² + bx + c
A standard method from Polynomials and factorisation: the steps and worked examples.
Steps
- List factor pairs of a (left column) and of c (right column).
- Cross-multiply and add. The sum must be the middle coefficient b.
- Read the brackets across the rows.
2x +3 2x × (−5) = −10x
╳ 3x × (+3) = +9x
3x −5 −x ✓ matches −xSo 6x² − x − 15 = (2x + 3)(3x − 5).
Tip on signs: if c is positive, both signs are the same as b's sign. If c is negative, the signs are different.
Worked examples
Example 1: Factorise 6x² − x − 15 by the cross-method.
Cross-method above: (2x + 3)(3x − 5).
Example 2: Factorise 3x² + 10x − 8 by the cross-method.
3x −2 3x × (+4) = +12x
╳ x × (−2) = −2x
x +4 +10x ✓ matches +10xc is negative, so the signs differ. 3x² + 10x − 8 = (3x − 2)(x + 4).
For the whole topic, read Polynomials and factorisation: key points, common mistakes and how the DSE tests it.