Three relationships for the cubic ax^3+bx^2+cx+d=0?

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

Four relationships for the quartic ax^4+bx^3+cx^2+dx+e=0?

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

Express \alpha^2+\beta^2+\gamma^2 using the relationships.

\left(\sum\alpha\right)^2-2\sum\alpha\beta.

Express \frac1\alpha+\frac1\beta+\frac1\gamma for a cubic.

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

Express \frac1{\alpha\beta}+\frac1{\beta\gamma}+\frac1{\gamma\alpha}.

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

How do you find an equation whose roots are 2\alpha+1 etc.?

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

Roots and coefficients of polynomialsAQA A-Level Further Maths: Flashcards

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Sum and product of roots of $ax^2+bx+c=0$?

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Sum and product of roots of $ax^2+bx+c=0$ ?: $\alpha+\beta=-\frac ba$ and $\alpha\beta=\frac ca$ .
Three relationships for the cubic $ax^3+bx^2+cx+d=0$ ?: $\sum\alpha=-\frac ba$ , $\sum\alpha\beta=\frac ca$ , $\alpha\beta\gamma=-\frac da$ .
Four relationships for the quartic $ax^4+bx^3+cx^2+dx+e=0$ ?: $\sum\alpha=-\frac ba$ , $\sum\alpha\beta=\frac ca$ , $\sum\alpha\beta\gamma=-\frac da$ , $\alpha\beta\gamma\delta=\frac ea$ .
Pattern of signs for a quartic?: They alternate: $-,+,-,+$ .
Express $\alpha^2+\beta^2+\gamma^2$ using the relationships.: $\left(\sum\alpha\right)^2-2\sum\alpha\beta$ .
Express $\frac1\alpha+\frac1\beta+\frac1\gamma$ for a cubic.: $\frac{\sum\alpha\beta}{\alpha\beta\gamma}$ .
Express $\frac1{\alpha\beta}+\frac1{\beta\gamma}+\frac1{\gamma\alpha}$ .: $\frac{\sum\alpha}{\alpha\beta\gamma}$ .
How do you find an equation whose roots are $2\alpha+1$ etc.?: Substitute $x=\frac{y-1}{2}$ into the original equation and tidy up.
A cubic has roots with sum $S$ , pairs $P$ and product $Q$ . Write its equation.: $y^3-Sy^2+Py-Q=0$ .
How do you write three roots in arithmetic progression?: $a-d$ , $a$ , $a+d$ .
How do you write three roots in geometric progression?: $\frac ar$ , $a$ , $ar$ .
If every root of a cubic is multiplied by 3, how does $\sum\alpha\beta$ change?: It is multiplied by $3^2=9$ .

Exam questions on Roots and coefficients of polynomials

The roots of the equation $x^3-4x^2+5x-7=0$ are $\alpha$ , $\beta$ and $\gamma$ .
Find the value of $\frac1\alpha+\frac1\beta+\frac1\gamma$ .2 marks
The roots of the equation $x^4-6x^3+3x^2+2x-5=0$ are $\alpha$ , $\beta$ , $\gamma$ and $\delta$ .
Find the value of $\alpha^2+\beta^2+\gamma^2+\delta^2$ .2 marks
The roots of the equation $x^3+3x^2-5x+1=0$ are $\alpha$ , $\beta$ and $\gamma$ .
Show that $\alpha^2+\beta^2+\gamma^2=19$ .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).