Find one root, factor it out, repeat: Spot one root by trying small values (the factor theorem). Divide it out to get a quadratic, then factorise or solve that. The roots fall straight out.
IB-style question — factorise a cubic
Factorise x³ − 2x² − 5x + 6 fully and state its roots.
Step by step
- Try x = 1: P(1) = 1 − 2 − 5 + 6 = 0, so (x − 1) is a factor.
- Divide out (x − 1).
- Factorise the quadratic.
Final answer
Roots x = 1, 3, −2.
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Complex roots come in pairs — use the sum: If the coefficients are real and a + bi is a root, so is a − bi (from Unit 1.14). The sum of all roots (= −b/a) then hands you the remaining real root.
IB-style question — find the other roots
Given that 1 + i is a root of x³ − 4x² + 6x − 4 = 0, find the other two roots.
Step by step
- Real coefficients ⇒ the conjugate is also a root.
- Sum of all three roots = −b/a = 4.
- The two complex roots add to 2, so the real root is 4 − 2.
Final answer
The other roots are 1 − i and 2.