Question

If alpha, beta, gamma, delta are the roots of the equation x^4+ax^3+bx^2+cx+d=0,find the value of Sigma alpha square beta

Answer 1

Answer:

 $\alpha ^2+\beta ^2+\gamma ^2+\delta ^2=a^2-2b$

Step-by-step explanation:

     Given that

$x^4+ax^3+bx^2+cx+d=0$

This is the 4th order equation .

We have to find

$\sum \alpha ^2=\alpha ^2+\beta ^2+\gamma ^2+\delta ^2$

As we know that

$\left ( a+b+c +d\right )^2=a^2+b^2+c^2+2\left ( ab+bc+ac+ad+db+cd \right )$

α+β+γ+δ= -a

αβ+βγ+γδ+δα+βδ+αγ= b

So

$\alpha ^2+\beta ^2+\gamma ^2+\delta ^2=\left ( \alpha +\beta +\gamma +\delta \right )^2-2\left ( \alpha \beta +\beta \gamma +\gamma \delta +\delta \alpha+\alpha \gamma +\beta \delta \right )$

Now by putting the values

$\alpha ^2+\beta ^2+\gamma ^2+\delta ^2=a^2-2b$

So

$\alpha ^2+\beta ^2+\gamma ^2+\delta ^2=a^2-2b$