Question

Using the given zero, find all other zeros of f(x).

-2i is a zero of f(x) = x4 - 45x2 - 196

Answer 1

Answers

2i, 14i, -14i

2i, 7, -7

2i, 14, -14

2i, 7i, -7i

Answer 2

Well, complex roots always come in pairs. If you have (x-i), then you have to (x+i). That being said, you x = -2i, which meansing (x+2i) is one of the factors. Since you need a pair, this means that (x-2i) is also a factor. From here, there are several ways you can go about it really. If it works out, I think this is the easiet route: Knowing that (x-2i) and (x+2i) are factors, we can multiply it out and see what we get. Foiling this, I would get

x^2 + 4 (feel free to ask if you want to see how I got that)

Now that I have x^2 + 4, its possible to try and find the other factors by polynomial long division. I would do this division:

\[\frac{ x ^{4}-45x ^{2}-196 }{ x ^{2}+4 } \]So now I can work out that long division:



Okay, awesome, we came up with x^2-49. Now this is a difference of squares that can be factored into (x+7)(x-7) *again, ask if you want to see how*. So this means our 4 roots in total are 2i, -2i, 7, -7.