Key Idea: This topic finds factors, remainders and roots of polynomials without long division. You also read the sum and product of roots straight from the coefficients. It is on Paper 1, Paper 2 and Paper 3.
| You are told | Write down |
|---|---|
| Remainder when dividing by (x − a) | P(a) |
| (x − a) is a factor | P(a) = 0 |
| (x − a)² is a factor | P(a) = 0 and P′(a) = 0 |
Sum & product of roots
- Quadratic ax² + bx + c: sum = −b/a, product = c/a.
- General: sum = −aₙ₋₁/aₙ, product = (−1)ⁿ a₀/aₙ.
- Use them to find an unknown without solving.
Finding all the roots & sketching
- One root — Test ±1, ±2 and factors of the constant.
- Divide out — Factorise or solve what is left.
- Complex root — Its conjugate is a root too. The sum of roots gives the last one.
- Sketch — Cross the x-axis at the real roots, y-intercept P(0), ends set by the leading term.
Each fact you are given is one equation, so match equations to unknowns.
x⁴ + ax + b has the factor (x + 1)². Find a and b. P′(−1) = −4 + a = 0, so a = 4. P(−1) = 1 − 4 + b = 0, so b = 3.
Roots of x² − kx + (k + 3) = 0 add to 5. Find k. Sum = k, so k = 5, and the product is 8.
Is the product of roots of a cubic + or − (constant)/(leading)? Minus, because (−1)³ = −1.
Roots of x³ + 3x² + 2x? 0, −1 and −2.