Write \alpha^3+\beta^3 in terms of \alpha+\beta and \alpha\beta.

(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)

Write \frac1\alpha+\frac1\beta in terms of \alpha+\beta and \alpha\beta.

\frac{\alpha+\beta}{\alpha\beta}

Write \alpha^4+\beta^4 in terms of \alpha^2+\beta^2 and \alpha\beta.

(\alpha^2+\beta^2)^2-2(\alpha\beta)^2

Write \frac{\alpha}{\beta}+\frac{\beta}{\alpha} using the sum and product.

\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}

What is a symmetric expression in \alpha and \beta?

One that is unchanged when \alpha and \beta are swapped, so it can be written using \alpha+\beta and \alpha\beta.

Why do we not solve the quadratic in these questions?

The roots may be irrational or complex, and the sum and product give the answer exactly and faster.

Manipulating expressions in the rootsEdexcel International A Level Further Maths: Flashcards

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Question

For $ax^2+bx+c=0$, what is $\alpha+\beta$?

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For $ax^2+bx+c=0$ , what is $\alpha+\beta$ ?: $-\frac{b}{a}$
For $ax^2+bx+c=0$ , what is $\alpha\beta$ ?: $\frac{c}{a}$
Write $\alpha^2+\beta^2$ in terms of $\alpha+\beta$ and $\alpha\beta$ .: $(\alpha+\beta)^2-2\alpha\beta$
Write $\alpha^3+\beta^3$ in terms of $\alpha+\beta$ and $\alpha\beta$ .: $(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$
Write $\frac1\alpha+\frac1\beta$ in terms of $\alpha+\beta$ and $\alpha\beta$ .: $\frac{\alpha+\beta}{\alpha\beta}$
Write $(\alpha-\beta)^2$ using the sum and product.: $(\alpha+\beta)^2-4\alpha\beta$
Write $\alpha^4+\beta^4$ in terms of $\alpha^2+\beta^2$ and $\alpha\beta$ .: $(\alpha^2+\beta^2)^2-2(\alpha\beta)^2$
Write $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$ using the sum and product.: $\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}$
Write $(\alpha+2)(\beta+2)$ using the sum and product.: $\alpha\beta+2(\alpha+\beta)+4$
What is a symmetric expression in $\alpha$ and $\beta$ ?: One that is unchanged when $\alpha$ and $\beta$ are swapped, so it can be written using $\alpha+\beta$ and $\alpha\beta$ .
$x^2-6x+4=0$ : find $\alpha^2+\beta^2$ .: $36-8=28$
Why do we not solve the quadratic in these questions?: The roots may be irrational or complex, and the sum and product give the answer exactly and faster.

Exam questions on Manipulating expressions in the roots

The roots of the equation $x^2-6x+4=0$ are $\alpha$ and $\beta$ .
Find the value of $(\alpha+1)(\beta+1)$ .2 marks
The roots of the equation $2x^2+5x-3=0$ are $\alpha$ and $\beta$ .
Find the exact value of $(\alpha-\beta)^2$ .2 marks
The roots of the equation $x^2-5x+2=0$ are $\alpha$ and $\beta$ .
Find the value of $\alpha^3+\beta^3$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).