| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Roots of polynomials |
| Type | Substitution to find new equation |
| Difficulty | Challenging +1.2 This is a standard Further Maths question on polynomial transformations and recurrence relations. Part (i) requires the routine substitution y = 2x + 1, then algebraic manipulation. Part (ii) uses Newton's sums or recurrence relations with the transformed equation. While it requires multiple techniques and careful algebra, these are well-practiced methods in Further Maths syllabi with no novel insight needed. Slightly above average difficulty due to the multi-step nature and Further Maths content. |
| Spec | 4.05b Transform equations: substitution for new roots |
8 The equation $8 x ^ { 3 } + 12 x ^ { 2 } + 4 x - 1 = 0$ has roots $\alpha , \beta , \gamma$.\\
(i) By considering a suitable substitution, or otherwise, show that the equation whose roots are $2 \alpha + 1,2 \beta + 1,2 \gamma + 1$ can be written in the form
$$y ^ { 3 } - y - 1 = 0 .$$
(ii) The sum $( 2 \alpha + 1 ) ^ { n } + ( 2 \beta + 1 ) ^ { n } + ( 2 \gamma + 1 ) ^ { n }$ is denoted by $S _ { n }$.
Evaluate $S _ { 3 }$ and $S _ { - 2 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q8}}