CAIE A-Level Further Mathematics 1.1.2 Root Transformations

Practise substituting transformed roots into a polynomial, clearing powers or denominators and simplifying the resulting equation for shifted, reciprocal or squared roots.

Exam points

choose a substitution linking the old and new roots, then insert it into the original polynomial
clear fractions or radicals after substitution and simplify to the required polynomial equation

[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, $x=y^{\frac{1}{2}}$

B1

Correct substitution.

$y^{2}-y+2 y^{\frac{1}{2}}+5=0$ so, $\left(2 y^{\frac{1}{2}}\right)^{2}=\left(-y^{2}+y-5\right)^{2}$

M1

Obtains an equation which eliminates radicals.

$y^{4}-2 y^{3}+11 y^{2}-14 y+25=0$

A1

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

B1FT

4

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1.1.2—Root transformations question 1