Challenging +1.8 This question requires systematic application of Vieta's formulas and transformation of roots across two cubic equations. Students must relate coefficients through sums and products of roots, then use the linear transformation w = 4x - 1 to connect the two equations. While the techniques are standard A-level Further Maths content, the multi-step algebraic manipulation and coordination of three unknowns across two equations requires careful reasoning beyond routine exercises.
The cubic equation
$$ax^3 + bx^2 - 19x - b = 0$$
where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\)
The cubic equation
$$w^3 - 9w^2 - 97w + c = 0$$
where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\)
Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
[6]
The cubic equation
$$ax^3 + bx^2 - 19x - b = 0$$
where $a$ and $b$ are constants, has roots $\alpha$, $\beta$ and $\gamma$
The cubic equation
$$w^3 - 9w^2 - 97w + c = 0$$
where $c$ is a constant, has roots $(4\alpha - 1)$, $(4\beta - 1)$ and $(4\gamma - 1)$
Without solving either cubic equation, determine the value of $a$, the value of $b$ and the value of $c$.
[6]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q8 [6]}}