| Exam Board | SPS |
|---|---|
| Module | SPS ASFM (SPS ASFM) |
| Year | 2020 |
| Session | May |
| Marks | 9 |
| Topic | Roots of polynomials |
| Type | Equation with nonlinearly transformed roots |
| Difficulty | Standard +0.8 This question requires systematic application of Vieta's formulas and algebraic manipulation to find symmetric functions of roots, then construct a new polynomial. Part (a) needs expressing α²β² + β²γ² + γ²α² in terms of elementary symmetric functions, requiring the identity (αβ + βγ + γα)² - 2αβγ(α + β + γ). Part (b) requires finding sum, sum of products, and product of squared roots to construct the new cubic. While the techniques are standard A-level further maths content, the multi-step algebraic manipulation and careful bookkeeping with fractions makes this moderately challenging. |
| Spec | 4.05a Roots and coefficients: symmetric functions |
\textbf{In this question you must show detailed reasoning.}
You are given that $\alpha$, $\beta$ and $\gamma$ are the roots of the equation $5x^3 - 2x^2 + 3x + 1 = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2$. [5]
\item Find a cubic equation whose roots are $\alpha^2$, $\beta^2$ and $\gamma^2$ giving your answer in the form $ax^3 + bx^2 + cx + d = 0$ where $a$, $b$, $c$ and $d$ are integers. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM 2020 Q4 [9]}}