Quartic ax^4+bx^3+cx^2+dx+e=0: the four symmetric sums?

\sum\alpha=-\frac ba, \sum\alpha\beta=\frac ca, \sum\alpha\beta\gamma=-\frac da, \alpha\beta\gamma\delta=\frac ea

Formula for \alpha^2+\beta^2+\gamma^2?

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

Formula for \frac1\alpha+\frac1\beta+\frac1\gamma of a cubic?

\frac{\sum\alpha\beta}{\alpha\beta\gamma}

Expand (k+\alpha)(k+\beta)(k+\gamma) in symmetric sums.

k^3+k^2\sum\alpha+k\sum\alpha\beta+\alpha\beta\gamma

Two methods for \alpha^3+\beta^3+\gamma^3?

Use the identity (\sum\alpha)^3-3\sum\alpha\sum\alpha\beta+3\alpha\beta\gamma, or substitute each root into the equation and sum.

What does a negative \sum\alpha^2 tell you?

At least one root is not real, because real roots have non-negative squares.

How do you form an equation with roots 2\alpha+1?

Substitute x=\frac{y-1}{2} into the original equation and clear fractions.

How do you form an equation with roots \frac1\alpha?

Substitute x=\frac1y and multiply by the highest power of y; the coefficients reverse.

Quartic: the product \alpha\beta\gamma\delta in terms of a and e?

+\frac ea, since the signs alternate -,+,-,+.

Roots of polynomialsEdexcel A-Level Further Maths: Flashcards

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Cubic $ax^3+bx^2+cx+d=0$: $\sum\alpha$, $\sum\alpha\beta$, $\alpha\beta\gamma$?

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Cubic $ax^3+bx^2+cx+d=0$ : $\sum\alpha$ , $\sum\alpha\beta$ , $\alpha\beta\gamma$ ?: $-\frac ba$ , $\frac ca$ , $-\frac da$
Quartic $ax^4+bx^3+cx^2+dx+e=0$ : the four symmetric sums?: $\sum\alpha=-\frac ba$ , $\sum\alpha\beta=\frac ca$ , $\sum\alpha\beta\gamma=-\frac da$ , $\alpha\beta\gamma\delta=\frac ea$
Formula for $\alpha^2+\beta^2+\gamma^2$ ?: $(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)$
Formula for $\frac1\alpha+\frac1\beta+\frac1\gamma$ of a cubic?: $\frac{\sum\alpha\beta}{\alpha\beta\gamma}$
Expand $(k+\alpha)(k+\beta)(k+\gamma)$ in symmetric sums.: $k^3+k^2\sum\alpha+k\sum\alpha\beta+\alpha\beta\gamma$
Two methods for $\alpha^3+\beta^3+\gamma^3$ ?: Use the identity $(\sum\alpha)^3-3\sum\alpha\sum\alpha\beta+3\alpha\beta\gamma$ , or substitute each root into the equation and sum.
What does a negative $\sum\alpha^2$ tell you?: At least one root is not real, because real roots have non-negative squares.
How do you form an equation with roots $2\alpha+1$ ?: Substitute $x=\frac{y-1}{2}$ into the original equation and clear fractions.
How do you form an equation with roots $\frac1\alpha$ ?: Substitute $x=\frac1y$ and multiply by the highest power of $y$ ; the coefficients reverse.
New sum of roots when the roots become $p\alpha+q$ in a cubic?: $p\sum\alpha+3q$
Quartic: the product $\alpha\beta\gamma\delta$ in terms of $a$ and $e$ ?: $+\frac ea$ , since the signs alternate $-,+,-,+$ .
For $x^3-6x^2+11x-6=0$ (roots $1,2,3$ ), the equation with roots $2\alpha$ ?: $y^3-12y^2+44y-48=0$

Exam questions on Roots of polynomials

The cubic equation $2x^3-6x^2+3x+5=0$ has roots $\alpha$ , $\beta$ and $\gamma$ .
Find the value of $\frac1\alpha+\frac1\beta+\frac1\gamma$ .2 marks
The quartic equation $x^4+2x^3-7x^2+4x-3=0$ has roots $\alpha$ , $\beta$ , $\gamma$ and $\delta$ .
Find the value of $\alpha^2+\beta^2+\gamma^2+\delta^2$ .2 marks
The cubic equation $x^3+3x^2-5x+2=0$ has roots $\alpha$ , $\beta$ and $\gamma$ .
Find the value of $(3+\alpha)(3+\beta)(3+\gamma)$ .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).