Question

What are the possible integral zeros of `P(p)=p^4-2p^3-8p^2+3p-4`?

Answer 1

The "possible" integral zeros are: `+-1, +-2, +-4`

Actually `P(p)` has no rational zeros.

Explanation:

Given:

`P(p) = p^4-2p^3-8p^2+3p-4`

By the rational roots theorem, any rational zeros of `P(p)` are 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 (which also happen to be integers) are:

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

In practice we find that none of these are actually zeros, so `P(p)` has no rational zeros.