Factor x^3-3x^2+1

Problem

x3−3*x2+1

Solution

1. Test for rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term 1 divided by the factors of the leading coefficient 1 which are ±1

2. Evaluate the polynomial at x=1 1−3*(1)2+1=1−3+1=−1 Since this is not zero, x=1 is not a root.

3. Evaluate the polynomial at x=−1 (−1)3−3*(−1)2+1=−1−3+1=−3 Since this is not zero, x=−1 is not a root.

4. Conclude irreducibility over the rational numbers. Since the polynomial is of degree 3 and has no rational roots, it cannot be factored into polynomials with rational coefficients.

5. Determine the nature of roots using the discriminant for a cubic a*x3+b*x2+c*x+d The discriminant is Δ=18*a*b*c*d−4*b3*d+b2*c2−4*a*c3−27*a2*d2

6. Calculate the discriminant with a=1,b=−3,c=0,d=1 Δ=18*(1)*(−3)*(0)*(1)−4*(−3)3*(1)+(−3)2*(0)2−4*(1)*(0)3−27*(1)2*(1)2

7. Simplify the discriminant: Δ=0−4*(−27)+0−0−27=108−27=81 Since Δ>0 there are three distinct real roots, but they are irrational.

8. State the result that the expression is irreducible over the field of rational numbers ℚ

Final Answer

x3−3*x2+1=irreducible over *ℚ

Want more problems? Check here!