How to use the rational root theorem?

No, p(3)=913

factors of a0 coefficient 10 -->1,2,5

factors of the highest power x^5 a5=4 --> 1,2,4

We should check

1,2,5 / 1,2,4

We have

± 1,2,5,1/2,5/2, 1/4, 5/4

None of them plugged into the polynomial gives 0.

So we can state that teh polynomial

doesn't have RATIONAL roots.

But it has 5 non rational roots, some of them can be complex.

I found especially for you one root, not rational.

<<<<<<<<<<<<<<<<<<<

x0= - 1.5080375 .....

>>>>>>>>>>>>>>>>>>>>

p(x0)=8.81 *10^(-7) !

Rational root theorem says that you

a) take the constant term and divide by the first term

i.e., 10/4

b) Reduce it so that the fraction can't be reduced anymore

i.e. 5/2

Now all rational roots must have numerators that are integer factors of 5 (i.e., 1 and 5) and denominators that are integer factors of 2 (i.e., 2 and 1)

That means that rational roots can be 1/2, 5/2, 1, and 5. Since none of those are 3, that means f(3) is not zero because if f(3) = 0 then 3 would be a root.