Question

How do you use the rational roots theorem to find all possible zeros of `P(x) = 2x^5 + 3x^4 + 2x^2 - 2`?

Answer 1

We seek for rational roots within numbers `p/q`, where `p | a_0` and `q | a_n`.
If `p | -2` then `p in {+-1,+-2}`.
If `q | 2` then `q in {+-1,+-2}`.
Thus `p/q in R={+-1,+-1/2,+-2}`.

Now, if the polynomian `P(x)` has any rational roots, they all are in `R` so we proceed by trial and error:

`P(1)=5 !=0`
`P(-1)=1 !=0`
`P(0.5)=-1.25 !=0`
`P(-0.5)=-1.375 !=0`
`P(2)=118 !=0`
`P(-2)=-10 !=0`

From this we can conclude that the polynomial `P(x)` has no rational roots.