CAIE A-Level Further Math ASematics 1.1.1 Polynomial Roots

[Maximum number: 8]

The quartic equation $x^{4}+b x^{3}+c x^{2}+d x-2=0$ has roots $\alpha, \beta, \gamma, \delta$ . It is given that

\alpha+\beta+\gamma+\delta=3, \quad \alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}=5, \quad \quad \alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1}=6 .

(a)

Find the values of b, c and d.

[ 6 ]

$b=-(\alpha+\beta+\gamma+\delta)=-3$ $5=(-3)^{2}-2(\alpha \beta+\alpha \gamma+\alpha \delta+\beta \gamma+\beta \delta+\gamma \delta)$
Uses formula for sum of squares.

c=2
$6=\frac{\alpha \beta \gamma+\beta \gamma \delta+\gamma \delta \alpha+\delta \alpha \beta}{\alpha \beta \gamma \delta}=\frac{-d}{-2}$
Uses
$\alpha^{-1}+\beta^{-1}+\gamma^{-1}+\delta^{-1}=\frac{\alpha \beta \gamma+\beta \gamma \delta+\gamma \delta \alpha+\delta \alpha \beta}{\alpha \beta \gamma \delta}.$

d=12
Equation is $x^{4}-3 x^{3}+2 x^{2}+12 x-2=0$ .

6

(b)

Given also that $\alpha^{3}+\beta^{3}+\gamma^{3}+\delta^{3}=-27$ , find the value of $\alpha^{4}+\beta^{4}+\gamma^{4}+\delta^{4}$ .

[ 2 ]

$\alpha^{4}+\beta^{4}+\gamma^{4}+\delta^{4}=3(-27)-2(5)-12(3)+2(4)$
Uses their quartic equation derived in (a).

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CAIE A-Level Further Math AS 1.1.1 Polynomial Roots Questions

Exam points

CAIE A-Level Further Math AS 1.1.1 Polynomial Roots Questions question 1

Question (a)

Question (b)