Roots of Polynomials (Edexcel A Level Further Maths: Core Pure): Exam Questions

The roots of the equation

$x^3-2x^2+4x-5=0$

are $p,q$ and $r$.

Without solving the equation, find the value of

(i)   $\frac{2}{p}+\frac{2}{q}+\frac{2}{r}$

(ii)   $(p–4)(q–4)(r–4)$

(iii)   $p^3+q^3+r^3$

The roots of the equation

$x^3-8x^2+28x-32=0$

are $\alpha ,\beta$ and $\gamma$

Without solving the equation, find the value of

(i)   $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }$

(ii)   $(\alpha +2)(\beta +2)(\gamma +2)$

(iii)   ${\alpha }^2+{\beta }^2+{\gamma }^2$

The roots of the equation

$2x^3+5x^2-4x+3=0$

are $\alpha$, $\beta$ and $\gamma$.

Without solving the equation, write down the value of each of

$\alpha +\beta +\gamma \alpha \beta +\beta \gamma +\gamma \alpha \alpha \beta \gamma$

Use the answers to part (a) to determine the value of

(i) ${\alpha }^2+{\beta }^2+{\gamma }^2$

(ii) Hence, or otherwise, ${\alpha }^3+{\beta }^3+{\gamma }^3$

(iii) ${\alpha }^2{\beta }^2+{\beta }^2{\gamma }^2+{\gamma }^2{\alpha }^2$