| Exam Board | SPS |
|---|---|
| Module | SPS ASFM Mechanics (SPS ASFM Mechanics) |
| Year | 2021 |
| Session | May |
| Marks | 7 |
| Topic | Roots of polynomials |
| Type | Finding polynomial from root properties |
| Difficulty | Challenging +1.3 This is a standard A-level Further Maths question on roots of polynomials requiring systematic application of Vieta's formulas and given identities. Part (i) is straightforward substitution once you identify the coefficients, while part (ii) requires forming a new cubic from symmetric functions of the cubed roots—a technique covered in FM syllabi but requiring careful algebraic manipulation across multiple steps. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
In this question you must show detailed reasoning.
The equation $x^3 + 3x^2 - 2x + 4 = 0$ has roots $\alpha$, $\beta$ and $\gamma$.
\begin{enumerate}[label=(\roman*)]
\item Using the identity $\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma$ find the value of $\alpha^3 + \beta^3 + \gamma^3$. [3]
\item Given that $\alpha^3\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112$ find a cubic equation whose roots are $\alpha^3$, $\beta^3$ and $\gamma^3$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM Mechanics 2021 Q1 [7]}}