This module covers how to use substitution to find factors and remainders of polynomials, usually cubics, and how to finish a factorisation by dividing. In Additional Mathematics (0606) it leads into solving cubic equations, curve sketching and later algebra. Check the specification for your examination year on the Cambridge subject page for the exact scope.
The skills are small and connect in a clear chain. That makes this one of the more learnable parts of algebra once the order is right.
What should I know before starting?
You need comfortable expansion of brackets, factorising quadratics, and solving two linear equations together. If a quadratic such as 2x² + 5x + 2 does not factorise quickly, revisit quadratic structure and discriminants first. Fraction arithmetic matters too, because roots like ½ appear often.
One orienting example
Take f(x) = x³ − 2x² − 5x + 6. Test x = 1: 1 − 2 − 5 + 6 = 0, so (x − 1) is a factor (factor theorem).
Divide: x³ − 2x² − 5x + 6 = (x − 1)(x² − x − 6). Factorise the quadratic: x² − x − 6 = (x − 3)(x + 2). The full factorisation is (x − 1)(x − 3)(x + 2).
Expand to check: (x − 1)(x − 3)(x + 2) = (x² − 4x + 3)(x + 2) = x³ − 2x² − 5x + 6. ✓ Every lesson in this module is one piece of that example.
In what order should I study the lessons?
- Use a known root to obtain a factor: the test-and-spot step that starts every cubic.
- Find a remainder without full division: the same substitution, now with a non-zero answer.
- Divide a cubic by a linear factor: turns one factor into a full factorisation.
- Reconstruct a polynomial from conditions: uses the first two lessons to find unknown coefficients.
- Check a factorisation by expansion: a quick habit that catches sign errors.
Then work through the mixed practice set. The non-calculator working trainer supports the arithmetic, and the quadratic structure explorer helps with the quadratic left after division.
What are the common traps?
- Wrong sign of the root. The factor (x + 2) means the root is −2, and (x − 3) means the root is 3.
- Substituting the number in the divisor. Set the divisor to zero and solve first.
- Skipping a missing power. A cubic with no x² term needs a 0 in the coefficient row.
- Treating a remainder as a factor. A remainder of 12 means f(a) = 12, not 0.
- Not checking. A thirty-second expansion catches most slips.
How should I use the practice set?
Attempt each question on paper before opening the solution. Mark where your working differs, and note the error type. The practice page routes each type of error back to the right lesson, so you revise the skill that failed and not the whole module.
When the same type of mistake repeats after revision, that usually needs someone watching your written working. Our teachers do this in online one-to-one Additional Mathematics tuition, and the wider Additional Mathematics learning guide shows where this module sits in the syllabus route. Once you are confident here, algebraic equations and inequalities is a natural next module.