| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | May |
| Marks | 7 |
| Topic | Roots of polynomials |
| Type | Sum of powers of roots |
| Difficulty | Standard +0.3 This is a straightforward application of Newton's identities/symmetric functions for roots of polynomials. Students need to use standard formulas (α+β+γ=-k, αβ+βγ+γα=0, αβγ=-9) and apply algebraic identities like (α+β+γ)²=α²+β²+γ²+2(αβ+βγ+γα). Both parts follow directly from these without requiring novel insight or complex manipulation, making it slightly easier than average for a Further Maths question. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
5.
The equation $z ^ { 3 } + k z ^ { 2 } + 9 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
(a)
\begin{enumerate}[label=(\roman*)]
\item Show that
$$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$
\item Show that
$$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q5 [7]}}