# Thread: Roots of polynomial equation

1. ## Roots of polynomial equation

Find the value of $\displaystyle \sum\alpha^2$, $\displaystyle \sum\alpha^2\beta^2$, $\displaystyle \sum\alpha\beta(\alpha+\beta)$, and $\displaystyle \sum\frac{1}{\alpha}$ for the following case:
$\displaystyle x^4-x^3+2x+3=0$

I started for the first one:
$\displaystyle \sum\alpha^2=\alpha^2+\beta^2+\gamma^2+\delta^2$
$\displaystyle =(\alpha+\beta+\gamma+\delta)^2-2(\alpha\beta+\beta\gamma+\gamma\delta+\alpha\delt a+\alpha\gamma+\beta\delta)$
$\displaystyle =(\sum\alpha)^2-2(\sum\alpha\beta)-2(\alpha\gamma+\beta\delta)$
This is as far as I got, the last term is what stumps me.
Thanks!

2. Originally Posted by arze
Find the value of $\displaystyle \sum\alpha^2$, $\displaystyle \sum\alpha^2\beta^2$, $\displaystyle \sum\alpha\beta(\alpha+\beta)$, and $\displaystyle \sum\frac{1}{\alpha}$ for the following case:
$\displaystyle x^4-x^3+2x+3=0$

I started for the first one:
$\displaystyle \sum\alpha^2=\alpha^2+\beta^2+\gamma^2+\delta^2$
$\displaystyle =(\alpha+\beta+\gamma+\delta)^2-2(\alpha\beta+\beta\gamma+\gamma\delta+\alpha\delt a+\alpha\gamma+\beta\delta)$
$\displaystyle =(\sum\alpha)^2-2(\sum\alpha\beta)-2(\alpha\gamma+\beta\delta)$
This is as far as I got, the last term is what stumps me.
Thanks!
In problems of this type, the usual convention is that $\displaystyle \sum\alpha\beta$ means the sum of all products of pairs of roots (not just products taken in cyclic order, but all pairs of products). In other words, $\displaystyle \sum\alpha\beta = \alpha\beta+\beta\gamma+\gamma\delta+\alpha\delta+ \alpha\gamma+\beta\delta$, and $\displaystyle \sum\alpha^2 = \bigl(\sum\alpha\bigr)^2 - 2\sum\alpha\beta$. With this convention, $\displaystyle \sum\alpha\beta$ is equal to the coefficient of $\displaystyle x^2$ in the quartic equation, so it is equal to 0.

3. For the second, I worked out the equivalent for cubic equations
$\displaystyle \sum\alpha^2\beta^2=(\sum\alpha\beta)^2-2\alpha\beta\gamma(\sum\alpha)$
Problem is it doesn't work.
$\displaystyle (0)^2-2(3)(-1)=6$ answer is 10.
Thanks!

4. Hello, arze!

The problem is not stated clearly.
And those Greek letters are too hard to type.

I'll do a few of these . . .

$\displaystyle \text}The roots of: }\,x^4 - x^3 + 2x + 3 \:=\:0\,\text{ are }\,a,b,c,d.$

$\displaystyle \text{Find the value of:}$

. . $\displaystyle (A)\;\;\sum\alpha^2$

. . $\displaystyle (B)\;\;\sum\alpha^2\beta^2$

. . $\displaystyle (C)\;\;\sum\alpha\beta(\alpha+\beta)$

. . $\displaystyle (D)\;\;\sum\frac{1}{\alpha}$

From Vieta's Theorem, we have these equations:

. . $\displaystyle \begin{array}{cccccc}a + b + c + d \;=\; 1 & [1] \\ ab + ac + ad + bc + bc + cd \;=\; 0 & [2] \\ abc + abd + acd + bcd \;=\; -2 & [3] \\ abcd \;=\; 3 & [4] \end{array}$

$\displaystyle (A)\;\text{ We want: }\:a^2 + b^2 + c^2 + d^2$

Square [1]: .$\displaystyle (a+b+c+d)^2 \:=\:1^2$

. . $\displaystyle a^2 + 2ab + 2ac + 2ad + b^2 + 2bc + 2bd + c^2 + 2cd + d^2 \;=\;1$

. . $\displaystyle a^2 + b^2 + c^2 + d^2 + 2\underbrace{(ab + ac + ad + bc + bd + cd)}_{\text{This is 0}} \;=\;1$

Therefore: .$\displaystyle a^2+b^2+c^2+d^2 \;=\;1$

$\displaystyle (D)\;\text{ We want: }\:\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} + \dfrac{1}{d}$

. . $\displaystyle \displaystyle \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \;=\;\frac{\overbrace{bcd + acd + abd + abc}^{\text{This is }-2}}{\underbrace{abcd}_{\text{This is 3}}}$

Therefore: .$\displaystyle \displaystyle \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \;=\;-\frac{2}{3}$

Get the idea?

5. Originally Posted by arze
For the second, I worked out the equivalent for cubic equations
$\displaystyle \sum\alpha^2\beta^2=(\sum\alpha\beta)^2-2\alpha\beta\gamma(\sum\alpha)$
Problem is it doesn't work.
$\displaystyle (0)^2-2(3)(-1)=6$ answer is 10.
Thanks!
You haven't got the formula for $\displaystyle \sum\alpha^2\beta^2$ quite right. It should be $\displaystyle \sum\alpha^2\beta^2=(\sum\alpha\beta)^2-2(\sum\alpha\beta\gamma)(\sum\alpha) +2\alpha\beta\gamma\delta = 0^2 -2(-2)(1) +2(3) = 10.$

6. Originally Posted by Opalg
You haven't got the formula for $\displaystyle \sum\alpha^2\beta^2$ quite right. It should be $\displaystyle \sum\alpha^2\beta^2=(\sum\alpha\beta)^2-2(\sum\alpha\beta\gamma)(\sum\alpha) +2\alpha\beta\gamma\delta = 0^2 -2(-2)(1) +2(3) = 10.$
So that formula would do for all quartic equations? Because I have another one, finding $\displaystyle \sum\alpha^2\beta^2$ for the equation $\displaystyle x^4+x-1=0$.
I should get
$\displaystyle (\sum\alpha\beta)^2-2(\sum\alpha\beta\gamma)(\sum\alpha)+2\alpha\btea\ gamma\delta=(o)^2-2(-1)(0)+2(-1)=-2$