Question

How do you find all the zeros of `f(x)=x^3+5x^2+x+5`?

Answer 1

`f(x)` has zeros `-5` and `+-i`

Explanation:

Since the ratio of the first and second terms is the same as that between the third and fourth terms, this cubic will factor by grouping.

So we find:

`x^3+5x^2+x+5`

`=x^2(x+5)+1(x+5)`

`=(x^2+1)(x+5)`

`=(x^2-i^2)(x+5)`

`=(x-i)(x+i)(x+5)`

Hence zeros: `+-i` and `-5`