Question

How do you use the rational roots theorem to find all possible zeros of `f(x)=x^4-x-4`?

Answer 1

The rational root theorem helps us determine that this `f(x)` has no rational zeros, only irrational and/or Complex ones.

Explanation:

`f(x) = x^4-x-4`

By the rational roots theorem, any rational zeros of `f(x)` must be expressible in the form `p/q` for integers `p`, `q` with `p` a divisor of the constant term `-4` and `q` a divisor of the coefficient `1` of the leading term.

That means that the only possible rational zeros are:

`+-1`, `+-2`, `+-4`

Trying each in turn, we find:

`f(1) = 1-1-4 = -4`

`f(-1) = 1+1-4 = -2`

`f(2) = 16-2-4 = 10`

`f(-2) = 16+2-4 = 14`

`f(4) = 256-4-4 = 248`

`f(-4) = 256+4-4 = 256`

So there are no rational zeros, but `f(x)` changes sign in `(-2, -1)` and `(1, 2)`, so there are irrational zeros in those intervals.

That's as much as we can learn about this `f(x)` from the rational root theorem.