Challenging +1.2 This is a Further Maths question on transformed roots requiring systematic application of Vieta's formulas and algebraic manipulation. While it involves multiple steps (finding sums of roots, products, and constructing the new equation), the technique is standard for FM students who have practiced this topic. The transformation (pairwise sums) is straightforward compared to more complex transformations like reciprocals or squares, making it moderately above average difficulty.
3 In this question you must show detailed reasoning.
The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) 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 roots of the equation $4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0$ are $\alpha , \beta$ and $\gamma$.
Find a cubic equation with integer coefficients whose roots are $\alpha + \beta , \beta + \gamma$ and $\gamma + \alpha$.
\hfill \mbox{\textit{OCR Further Pure Core 2 2022 Q3 [6]}}