The cubic equation \(x^3 + px^2 + qx + r = 0\), where \(p\), \(q\) and \(r\) are integers, has roots \(\alpha\), \(\beta\) and \(\gamma\), such that $$\alpha + \beta + \gamma = 15,$$ $$\alpha^2 + \beta^2 + \gamma^2 = 83.$$ Write down the value of \(p\) and find the value of \(q\). [3] Given that \(\alpha\), \(\beta\) and \(\gamma\) are all real and that \(\alpha\beta + \alpha\gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]

Question 5:
5 | α +β +γ =−p=15⇒ p=−15
( )
2(αβ +βγ +γα)=(α +β +γ)2− α2+β2+γ2 =2q
1
⇒q= (225−83)=71
2
36
=15−α (=[β +γ])
α
⇒a2 −15α +36=0⇒α =3, α ≠12, e.g. since 12 2 > 83 or other reason
βγ =71−36=35
⇒r=−αβγ =−3×35=−105 (extra answer penalised) | B1
M1
A1
[3]
M1
M1A1
B1
A 1
[5]
Total
8

The cubic equation $x^3 + px^2 + qx + r = 0$, where $p$, $q$ and $r$ are integers, has roots $\alpha$, $\beta$ and $\gamma$, such that
$$\alpha + \beta + \gamma = 15,$$
$$\alpha^2 + \beta^2 + \gamma^2 = 83.$$

Write down the value of $p$ and find the value of $q$. [3]

Given that $\alpha$, $\beta$ and $\gamma$ are all real and that $\alpha\beta + \alpha\gamma = 36$, find $\alpha$ and hence find the value of $r$. [5]

\hfill \mbox{\textit{CAIE FP1 2015 Q5 [8]}}