Find the complex zeros of the following polynomial function. Write f in factored form. f(x) = x⁴ + 26x² + 25

Solution:

Using the u- substitution in the given polynomial we have,

Let u = x²

f(u) = u² + 26u + 25 is a quadratic equation. By splitting middle terms we get

f(u) =  u² + 25u + u  + 25

=  u(u + 25) + 1(u + 25)

=  (u + 1)(u + 25)

To find the zeros of the above equation we put,

f(u) = (u +1)(u + 25) = 0

u = -1  and u = -25

Since u =  x² we get the solution for x as 

 x² = -1  ⇒  x = ± i

Also u = -25 which implies 

x² = -25   ⇒   x = ± 5 i

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Find the complex zeros of the following polynomial function. Write f in factored form.

Summary:

The complex zeros of the following polynomial function f(x) = x⁴ + 26x² + 25 are x = ± i and x = ± 5 i.