CAIE A-Level Further Math AS 1.1.2 Root Transformations Questions

Practise transforming polynomial roots and forming equations for transformed roots.

Exam points

form equations with transformed polynomial roots
calculate sums of powers for transformed roots
solve root transformation problems in context

[Maximum number: 4]

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

Find a quartic equation whose roots are $\alpha^{2}, \beta^{2}, \gamma^{2}, \delta^{2}$ and state the value of $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$ .

$y=x^{2}$ so, $x=y^{\frac{1}{2}}$
Correct substitution.

$y^{2}-y+2 y^{\frac{1}{2}}+5=0$ so, $\left(2 y^{\frac{1}{2}}\right)^{2}=\left(-y^{2}+y-5\right)^{2}$
Obtains an equation which eliminates radicals.

$y^{4}-2 y^{3}+11 y^{2}-14 y+25=0$ $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}=2$

B1FT

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CAIE A-Level Further Math AS 1.1.2 Root Transformations Questions question 1