Challenging +1.2 This is a Further Maths question involving polynomial root relationships and recurrence relations. While it requires understanding of symmetric functions and the substitution y=x², the method is fairly standard for FP1: square the original equation, rearrange to eliminate odd powers, then use Newton's recurrence. The final part uses the new equation with S_n notation systematically. More challenging than typical A-level due to algebraic manipulation length, but follows a well-established technique taught in Further Maths courses.
2 The roots of the equation \(x ^ { 4 } - 4 x ^ { 2 } + 3 x - 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\); the sum \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). By using the relation \(y = x ^ { 2 }\), or otherwise, show that \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\) and \(\delta ^ { 2 }\) are the roots of the equation
$$y ^ { 4 } - 8 y ^ { 3 } + 12 y ^ { 2 } + 7 y + 4 = 0$$
State the value of \(S _ { 2 }\) and hence show that
$$S _ { 8 } = 8 S _ { 6 } - 12 S _ { 4 } - 72 .$$
2 The roots of the equation $x ^ { 4 } - 4 x ^ { 2 } + 3 x - 2 = 0$ are $\alpha , \beta , \gamma$ and $\delta$; the sum $\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }$ is denoted by $S _ { n }$. By using the relation $y = x ^ { 2 }$, or otherwise, show that $\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }$ and $\delta ^ { 2 }$ are the roots of the equation
$$y ^ { 4 } - 8 y ^ { 3 } + 12 y ^ { 2 } + 7 y + 4 = 0$$
State the value of $S _ { 2 }$ and hence show that
$$S _ { 8 } = 8 S _ { 6 } - 12 S _ { 4 } - 72 .$$
\hfill \mbox{\textit{CAIE FP1 2013 Q2 [6]}}