The factor theorem
- If f(a) = 0, then (x − a) is a factor of f(x). And if (x − a) is a factor, then f(a) = 0.
- For a factor (ax − b), substitute x = b/a. To show (2x − 1) is a factor, show f(1/2) = 0.
- Try small values (±1, ±2, ±3…) that divide the constant term.
Factorising a cubic
- f(x) = x³ − 6x² + 11x − 6. f(1) = 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor.
- Divide, or compare coefficients, to find the quadratic factor: x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6).
- Factorise the quadratic: (x − 1)(x − 2)(x − 3). So f(x) = 0 has solutions x = 1, 2 and 3.
- A cubic has at most three real roots.
Finding unknown coefficients
- If (x − 2) is a factor of x³ + ax² − 4x + 4, then f(2) = 0: 8 + 4a − 8 + 4 = 0, so 4a = −4 and a = −1.
- With two unknowns, use two facts to make simultaneous equations.
Key terms
- Factor theorem
- If f(a) = 0, then (x − a) is a factor of f(x).
- Cubic
- A polynomial with highest power x³.
- Root
- A value of x that makes f(x) = 0.
- Polynomial
- An expression with whole-number powers of x.
- Quadratic factor
- The quadratic left after taking a linear factor out of a cubic.