Question

How do you find all zeros of the function `f(x) = 3x^5 - x^2 + 2x + 18`?

Answer 1

Use a numerical method to find approximations for the zeros:

`x_1 ~~ -1.35047`

`x_(2,3) ~~ -0.506174+-1.35526i`

`x_(4,5) ~~ 1.18141+-0.852685i`

Explanation:

`f(x) = 3x^5-x^2+2x+18`

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `18` and `q` a divisor of the coefficient `3` of the leading term.

That means that the only possible rational zeros are:

`+-1/3, +-2/3, +-1, +-2, +-3, +-6, +-9, +-18`

None of these work, so `f(x)` has no rational zeros.

In common with most quintics and higher order polynomials, the zeros are not expressible in terms of `n`th roots or elementary functions, including trigonometric ones.

About the best you can do is use a numerical method like Durand-Kerner to find approximations:

`x_1 ~~ -1.35047`

`x_(2,3) ~~ -0.506174+-1.35526i`

`x_(4,5) ~~ 1.18141+-0.852685i`

See https://socratic.org/s/awNxzXZ9 for a description of the method and another example quintic approximated using this method.