Standard +0.8 This question requires understanding of symmetric functions of roots (Vieta's formulas) and sign analysis of polynomial roots. The first part is a standard manipulation using α+β+γ=8 and αβγ=-5, but the second part requires careful reasoning about polynomial behavior at specific points (f(0), sign changes) without graphing, which demands more mathematical maturity than typical A-level questions. This is appropriate for Further Maths but still accessible with systematic thinking.
4 The roots of the equation
$$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$
are \(\alpha , \beta , \gamma\). Show that
$$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$
It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.
4 The roots of the equation
$$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$
are $\alpha , \beta , \gamma$. Show that
$$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$
It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.
\hfill \mbox{\textit{CAIE FP1 2007 Q4 [7]}}