A-LevelFurther MathematicsRoots of polynomial equations9231/12 · Oct/Nov 2013 · Q26 Marks
The cubic equation x³ – px − q = 0, where p and q are constants, has roots α, β, γ. Show that
- (i)α² + β² + γ² = 2p,[1]
- (ii)α³ + β³ + γ³ = 3q,[2]
- (iii)6(α⁵ + β⁵ + γ⁵) = 5(α³ + β³ + γ³)(α² + β² + γ²).[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.
- [B1] (i) Finds S2. α² + β² + γ² = (α + β + γ)² - 2(αβ + βγ + γα) = 0 - 2(-p) = 2p (AG)
- [M1A1] (ii) Finds S3. α³ + β³ + γ³ = pΣα + 3q = 0 + 3q = 3q (AG)
- [M1] (iii) Finds S5. α⁵ + β⁵ + γ⁵ = pΣα³ + qΣα²
- [A1] = p.3q + q.2p = 5pq
- [A1] ⇒ 6Σα⁵ = 30pq = 5Σα³ Σα² (AG)
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