Factor x^3-5x^2+3x-2

Problem

x3−5*x2+3*x−2

Solution

1. Identify the polynomial as a cubic expression and check for integer roots using the Rational Root Theorem. The possible rational roots are factors of the constant term −2 divided by factors of the leading coefficient 1 which are ±1,±2

2. Test the possible roots by substituting them into the polynomial P(x)=x3−5*x2+3*x−2

P(1)=1−5+3−2=−3

P*(−1)=−1−5−3−2=−11

P(2)=8−20+6−2=−8

P*(−2)=−8−20−6−2=−36

3. Conclude that since none of the possible rational roots result in zero, the polynomial has no rational factors of the form (x−c)

4. Analyze the cubic further using the discriminant or numerical methods. The discriminant of a cubic a*x3+b*x2+c*x+d is given by Δ=18*a*b*c*d−4*b3*d+b2*c2−4*a*c3−27*a2*d2

Δ=18*(1)*(−5)*(3)*(−2)−4*(−5)3*(−2)+(−5)2*(3)2−4*(1)*(3)3−27*(1)2*(−2)2

Δ=540−1000+225−108−108=−451

5. Determine that because the discriminant is negative, the polynomial has one real root and two complex conjugate roots. Since the real root is not rational, the polynomial cannot be factored over the integers or rationals.

Final Answer

x3−5*x2+3*x−2=irreducible over the integers

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