Polynomials and Factorization
What the domain tests
Polynomial questions test degree, coefficients, zeroes, the remainder and factor theorems, and factorisation by splitting the middle term or using identities.
Core rules and concepts
- A polynomial of degree n has at most n real zeroes.
- Remainder theorem: the remainder when p(x) is divided by x − a is p(a).
- Factor theorem: x − a is a factor of p(x) exactly when p(a) = 0.
- For ax² + bx + c, the sum of zeroes is −b/a and the product is c/a.
- Common identities include a² − b² = (a − b)(a + b), and x² + (a + b)x + ab = (x + a)(x + b).
Worked example
Factorise p(x) = x³ − 4x² + x + 6. Test small integer values: p(2) = 8 − 16 + 2 + 6 = 0, so x − 2 is a factor. Divide by x − 2 to get x² − 2x − 3. Factorise the quadratic: x² − 2x − 3 = (x − 3)(x + 1). Hence p(x) = (x − 2)(x − 3)(x + 1), and the zeroes are 2, 3 and −1.
Traps that cost marks
A common error is to substitute a with the wrong sign: for x + 2, test p(−2), not p(2). When dividing, align powers of x and include missing terms with coefficient 0. A quadratic with no real zeroes can still be factorised over the rationals only if its discriminant is a perfect square; otherwise do not invent integer factors. Check each factor by expanding once.