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

Jan 14, 2017

The "possible" integral zeros are: $\pm 1 , \pm 2 , \pm 4$

Actually $P \left(p\right)$ has no rational zeros.

#### Explanation:

Given:

$P \left(p\right) = {p}^{4} - 2 {p}^{3} - 8 {p}^{2} + 3 p - 4$

By the rational roots theorem, any rational zeros of $P \left(p\right)$ are expressible in the form $\frac{p}{q}$ 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:

$\pm 1 , \pm 2 , \pm 4$

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