Prove that the polynomial f=x^3-3x^2+5x+1 has no integer roots.

According to the integer root theorem, the integer roots
of the given polynomial should be the factors of the constant term, namely
+1.

The factors of +1 are {-1 ;
+1}.

We'll calculate f(-1) and f(1) to verify if x = -1 and
x = 1 are the zeroes of the polynomial.

f(1) = 1^3 - 3*1^2
+ 5*1 + 1

f(1) = 1 - 3 + 5 +
1

f(1) = 4

We notice that x =
1 is not cancelling out the polynomial, therefore x = 1 is not a root for the given
polynomial.

We'll calculate
f(-1):

f(-1) = (-1)^3 - 3*(-1)^2 + 5*(-1) +
1

f(-1) = -1 - 3 - 5 + 1

f(-1)
= -8

We notice that x = -1 is not cancelling out the
polynomial, therefore x = -1 is not a root for the given
polynomial.

Since the factors of the constant
term are not the roots of the given polynomial, then there is no integer roots for the
polynomial f.