Standard +0.8 This is a Further Maths question requiring systematic use of Vieta's formulas to find relationships between roots of the original equation and products of pairs of roots in the new equation. While the technique is standard for FM students, it requires careful algebraic manipulation across multiple steps (finding sums and products of αβ, βγ, γα using symmetric functions) with significant scope for algebraic errors, placing it moderately above average difficulty.
6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
Find the values of \(p , q\) and \(r\).
Question 6:
6 | (cid:68)(cid:14)(cid:69)(cid:14)(cid:74)(cid:32)(cid:16)2, (cid:68)(cid:69)(cid:14)(cid:69)(cid:74)(cid:14)(cid:74)(cid:68)(cid:32)1, (cid:68)(cid:69)(cid:74)(cid:32)(cid:16)3
p =(cid:16)(cid:11)(cid:68)(cid:69)(cid:14)(cid:69)(cid:74)(cid:14)(cid:74)(cid:68)(cid:12)(cid:32)(cid:16)1
q(cid:32)(cid:68)(cid:69)(cid:69)(cid:74)(cid:14)(cid:69)(cid:74)(cid:74)(cid:68)(cid:14)(cid:74)(cid:68)(cid:68)(cid:69)
(cid:32)(cid:68)(cid:69)(cid:74)(cid:11)(cid:68)(cid:14)(cid:69)(cid:14)(cid:74)(cid:12)(cid:32)6
r(cid:32)(cid:16)(cid:68)(cid:69)(cid:69)(cid:74)(cid:74)(cid:68)(cid:32)(cid:16)9 | B1
B1
M1
A1
A1
[5] | 1.1
1.1
2.1
1.1
1.1 | soi
Manipulation of roots
6
n
e
m
i
c
6 The equation $x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.\\
The equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$ has roots $\alpha \beta , \beta \gamma$ and $\gamma \alpha$.\\
Find the values of $p , q$ and $r$.
\hfill \mbox{\textit{OCR Further Pure Core 1 Q6 [5]}}