A-LevelFurther MathematicsRoots of polynomial equations9231/12 · May/June 2016 · Q14 Marks
The roots of the cubic equation 2x³ + x² − 7 = 0 are α, β and γ. Using the substitution y = 1 + (1/x), or otherwise, find the cubic equation whose roots are 1 + (1/α), 1 + (1/β) and 1 + (1/γ), giving your answer in the form ay³ + by² + cy + d = 0, where a, b, c and d are constants to be found.
- The roots of the cubic equation 2x³ + x² − 7 = 0 are α, β and γ. Using the substitution y = 1 + (1/x), or otherwise, find the cubic equation whose roots are 1 + (1/α), 1 + (1/β) and 1 + (1/γ), giving your answer in the form ay³ + by² + cy + d = 0, where a, b, c and d are constants to be found.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [M1] y=1+1/x ⇒ x = 1/(y-1)
- [A1] 2/(y-1)³ + 1/(y-1)² - 7 = 0 ⇒ 2 + (y-1) - 7(y-1)³ = 0
- [M1A1] ⇒7(y³-3y²+3y-1)-y+1-2 = 0 ⇒ 7y³ - 21y² + 20y-8 = 0
- ALT METHOD: Σα, Σαβ, αβγM1 A1, ∑(1+1/a) etc M1 A1
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