A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2011 · Q26 Marks
The roots of the equation x³ + px² + qx + r = 0 are (β)/k , β , kβ , where p, q, r, k and β are non-zero real constants. Show that β = -q/p .
- The roots of the equation x³ + px² + qx + r = 0 are (β)/k , β , kβ , where p, q, r, k and β are non-zero real constants. Show that β = -q/p .[4]
- Deduce that \( rp^3 = q^3 \).[2]
Mark scheme
Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.
- [B1] Sum of roots. (β)/k + β + kβ = -p
- [B1] Sum of products in pairs. (β²)/k + kβ² + β² = q
- [M1] Factorises. ⇒ β ( (k²+k+1)/k ) = -p and β² ( (k²+k+1)/k ) = q
- [A1] ⇒ β = -q/p (AG)
- [B1] Product of roots. β³ = -r
- [B1] ⇒ -(q³)/(p³) = -r ⇒ rp³ = q³ (AG)
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