Standard +0.3 This is a standard Further Maths transformation of roots question requiring the substitution y = 1/x, leading to reversing coefficients. While it requires understanding of the technique and careful algebraic manipulation, it's a routine FP1 exercise with a well-known method that students are explicitly taught.
1 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
1 In this question you must show detailed reasoning.\\
The cubic equation $2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$. By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.
\hfill \mbox{\textit{OCR FP1 AS 2021 Q1 [3]}}