CAIE A-Level Further Math 1.1.1 Polynomial roots

Practise linking roots and coefficients in quadratic, cubic and quartic equations to evaluate symmetric sums and unknown constants.

[Maximum number: 5]

The equation $x^{4}-x^{2}+2 x+5=0$ has roots $\alpha, \beta, \gamma, \delta$ .

(a)

Find the value of $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}+\frac{1}{\delta^{2}}$ .

[ 3 ]

$\alpha^{2} \beta^{2} \gamma^{2} \delta^{2}=25$

B1FT

$\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}+\frac{1}{\delta^{2}}=\frac{\alpha^{2} \beta^{2} \delta^{2}+\alpha^{2} \beta^{2} \gamma^{2}+\beta^{2} \gamma^{2} \delta^{2}+\alpha^{2} \gamma^{2} \delta^{2}}{\alpha^{2} \beta^{2} \gamma^{2} \delta^{2}}$

M1

Relates to coefficients.

$\frac{14}{25}$

A1

CAO

3

(b)

Find the value of $\alpha^{4}+\beta^{4}+\gamma^{4}+\delta^{4}$ .

[ 2 ]

$\begin{aligned} \alpha^{4}+\beta^{4}+\gamma^{4}+\delta^{4}= \left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)^{2} \quad-2\left(\alpha^{2} \beta^{2}+\alpha^{2} \gamma^{2}+\alpha^{2} \delta^{2}+\beta^{2} \gamma^{2}+\beta^{2} \delta^{2}+\gamma^{2} \delta^{2}\right) = 2^{2}-2(11) \end{aligned}$

M1

Uses formula for sum of squares or uses original equation.

-18

A1

2

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1.1.1—Polynomial roots question 1

Question (a)

Question (b)