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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.