Question

What are all the possible rational zeros for `f(x)=x^5-6x^4+12x^2-8x+36` and how do you find all zeros?

Answer 1

Find there are no rational zeros. Use a numerical method to find approximations.

Explanation:

`f(x) = x^5-6x^4+12x^2-8x+36`

By the rational roots 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 `36` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1, +-2, +-3, +-4, +-6, +-9, +-12, +-18, +-36`

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

About the best we can do is use the Durand-Kerner or similar numerical method to find approximations:

`x_1 ~~ 5.63048`

`x_2 ~~ 2.05347`

`x_3 ~~ -1.84949`

`x_(4,5) ~~ 0.0827684+-1.29486i`

See https://socratic.org/s/axpvv6Hk for some more description of the method.

For the current example, the above approximations were found using this C++ program...