Standard +0.3 This is a standard Further Maths technique of finding a transformed cubic equation using substitution. Students substitute w = x + 3 (so x = w - 3) into the original equation and expand. While it requires careful algebraic manipulation, it's a routine procedure covered in FM Pure courses with no novel insight needed.
The cubic equation
$$2x^3 + 6x^2 - 3x + 12 = 0$$
has roots \(\alpha\), \(\beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \((\alpha + 3)\), \((\beta + 3)\) and \((\gamma + 3)\), giving your answer in the form \(pw^3 + qw^2 + rw + s = 0\), where \(p\), \(q\), \(r\) and \(s\) are integers to be found. [5]
The cubic equation
$$2x^3 + 6x^2 - 3x + 12 = 0$$
has roots $\alpha$, $\beta$ and $\gamma$.
Without solving the equation, find the cubic equation whose roots are $(\alpha + 3)$, $(\beta + 3)$ and $(\gamma + 3)$, giving your answer in the form $pw^3 + qw^2 + rw + s = 0$, where $p$, $q$, $r$ and $s$ are integers to be found. [5]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q4 [5]}}