Question

How do you find all zeros of the function `f(x)= x^4 + 5x^2 + 4` given -i as a zero?

Answer 1

Factor by grouping to find:

`x = +-i` or `x = +-2i`

Explanation:

This polynomial factors by grouping as follows:

`f(x) = x^4+5x^2+4`

`= x^4+4x^2+x^2+4`

`= x^2(x^2+4)+1(x^2+4)`

`= (x^2+1)(x^2+4)`

`= (x^2-i^2)(x^2-(2i)^2)`

`= (x-i)(x+i)(x-2i)(x+2i)`

Hence zeros: `x = +-i` and `x = +-2i`

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Another way of finding this is to note that since `f(x)` has Real coefficients, any zeros must occur in Complex conjugate pairs.

So if `x = -i` is a zero then so is `x = i`

Therefore `f(x)` is divisible by `(x+i)(x-i) = x^2-i^2 = x^2+1`

Then dividing `f(x)` by `x^2+1` we find:

`f(x) = (x^2+1)(x^2+4)`

Hence the other two zeros are `+-2i`