Moderate -0.3 This is a standard transformation of roots question requiring the substitution x = y/3 to scale all roots by factor 3, then clearing fractions. While it's a Further Maths topic, the technique is routine and mechanical with no problem-solving required—slightly easier than an average A-level question due to its straightforward algorithmic nature.
The cubic equation
$$x^3 + 5x^2 - 4x + 2 = 0$$
has roots \(\alpha\), \(\beta\) and \(\gamma\)
Find a cubic equation, with integer coefficients, whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\)
[3 marks]
Question 6:
6 | Sets y = 3x
PI by correct substitution
or
Obtains one of
∑ 3α= ±15
∑( 3α)( 3β)= ±36
( 3α)( 3β)( 3γ)= ±54 | 1.1a | M1 | Let y = 3x
y
Then x =
3
y3 5y2 4y
+ – +2=0
27 9 3
y3 + 15y2 – 36y + 54 = 0
y
Replaces x with or 3y
3
Accept x for y
or
Obtains at least two of
∑ 3α= ±15
∑( 3α)( 3β)= ±36
( 3α)( 3β)( 3γ)= ±54 | 1.1a | M1
Obtains y3 + 15y2 – 36y + 54 = 0
OE with integer coefficients. | 1.1b | A1
Question total | 3
Q | Marking instructions | AO | Marks | Typical solution
The cubic equation
$$x^3 + 5x^2 - 4x + 2 = 0$$
has roots $\alpha$, $\beta$ and $\gamma$
Find a cubic equation, with integer coefficients, whose roots are $3\alpha$, $3\beta$ and $3\gamma$
[3 marks]
\hfill \mbox{\textit{AQA Further Paper 2 2024 Q6 [3]}}