Question

How do you find all the zeros of `x^5+x^3-30x`?

Answer 1

Factor to find zeros:

`0, +-sqrt(5)` and `+-sqrt(6)i`

Explanation:

After separating out the common factor `x`, the remaining quartic can be factored like a quadratic, before reducing to linear factors:

`x^5+x^3-30x = x(x^4+x^2-30)`

`color(white)(x^5+x^3-30x) = x(x^2+6)(x^2-5)`

`color(white)(x^5+x^3-30x) = x(x-sqrt(6)i)(x+sqrt(6)i)(x-sqrt(5))(x+sqrt(5))`

Hence zeros:

`0, +-sqrt(5)` and `+-sqrt(6)i`