# Find all complex zeros for the following polynomial function. f(x)= x^4 -9x^2 -400Please show all of your work.

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We will start by factoring the function

x^4-9x^2-400=(x^2-25)(x^2+16) If you set = 0

We will get (x^2-25)=0 or (x^2+16)=0, the first will give you 5 and -5 as answers, both real numbers but the second will lead to x^2=-16.

x^2=-16 ==> x=+ or - sqrt(-16)

**Therefore your answer will be 4i or -4i.**

You should come up with the substitution `x^2 = y` , hence you should write the equation in terms of y such that:

`y^2 - 9y - 400 = 0`

You should use quadratic formula to find the roots such that:

`y_(1,2) = (9+-sqrt(81 + 1600))/2 =gt y_(1,2) = (9+-sqrt1681)/2`

`` `y_(1,2) = (9+-41)/2 =gt y_1 = 25; y_2 = -16`

You need to solve for x the equations:

`x^2 = y_1 =gt x^2 = 25 =gt x_1=5; x_2 = -5 `

`x^2 = y_1 =gt x^2 = -16 =gt x_3 = 4i; x_4 = -4i ` (use complex number theory `sqrt(-1) = i` )

**Hence, the complex solutions to the polynomial equation are `x_3 = 4i; x_4 = -4i` .**