Challenging +1.8 This is a challenging Further Maths question requiring systematic use of Vieta's formulas to find sums and products of transformed roots, then constructing a new polynomial. While the technique is standard for FP1, the algebraic manipulation is substantial and error-prone, requiring careful tracking of symmetric functions through multiple steps—significantly harder than routine A-level questions but within reach of well-prepared FM students.
3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
3 In this question you must show detailed reasoning.\\
The cubic equation $5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
Find a cubic equation with integer coefficients whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.
\hfill \mbox{\textit{OCR FP1 AS 2021 Q3 [7]}}