Question

Find all zeros of a biquadratic polynomial if a+c+e equals 0 and b+d equals 0.

Answer 1

Answer:

Let the biquadratic equation be

$ax^4+bx^3+cx^2+dx+e$

given:

a+c+e=0 and b+d=0

That is,

sum of the coefficients of even powers of x = sum of the coefficients of odd powers of x

Hence, (x+1) is a factor

Also,

a+b+c+d+e=0

that is ,

sum of all the coefficients =0

Hence (x-1) is a factor

so, we have two zeros 1 and -1

Let the other two roots be $\alpha\:and\:\beta$

Then,

$S_1=\alpha+\beta+1+(-1)=\frac{-b}{a}$

$\alpha+\beta=\frac{-b}{a}$

$S_4=\alpha.\beta(1)(-1)=\frac{e}{a}$

$\alpha\beta=\frac{-e}{a}$

The required quadratic factor is

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

$x^2-(\frac{-b}{a})x+(\frac{-e}{a})$

$x^2+\frac{b}{a}x-\frac{e}{a}$

$\frac{1}{a}[ax^2+bx-e]$

The other two zeros are obtained by solving this quadratic polynomial.