Question

How to find imaginary zeros anf real zeros of F(x)=-4x^5-8x^3+12x​

Answer 1

`x=\{0, -1, 1, -i√(3), i√(3)\}`

Explanation:

We are given the function:

`f(x)=-4x^5-8x^3+12x`

And we want to finds its zeros.

Therefore:

`0=-4x^5-8x^3+12x`

Firstly, we can divide everything by -4:

`0=x^5+2x^3-3x`

Factor out an x:

`0=x(x^4+2x^2-3)`

This is in quadratic form. For simplicity, we can let:

`u=x^2`

Then by substitution:

`0=x(u^2+2u-3)`

Factor:

`0=x(u+3)(u-1)`

Substitute back:

`0=x(x^2+3)(x^2-1)`

By the Zero Product Property:

`x=0\text{ and } x^2+3=0\text{ and } x^2-1=0`

Solving for each case:

`x=0\text{ and } x=\pm√(-3)\text{ and } x=\pm√(1)`

Therefore, our real and complex zeros are:

`x=\{0, -1, 1, -i√(3), i√(3)\}`