Factor x^3-3x^2+12x-10

Problem

x3−3*x2+12*x−10

Solution

1. Identify a possible root using the Rational Root Theorem. Testing x=1 we find 1−3*(1)2+12*(1)−10=1−3+12−10=0

2. Apply the Factor Theorem, which states that since x=1 is a root, (x−1) is a factor of the polynomial.

3. Divide the polynomial x3−3*x2+12*x−10 by (x−1) using synthetic division or long division to find the remaining quadratic factor.

4. Calculate the quotient of the division, which results in x2−2*x+10

5. Check if the quadratic factor x2−2*x+10 can be factored further over the real numbers by examining the discriminant.

6. Evaluate the discriminant D=b2−4*a*c=(−2)2−4*(1)*(10)=4−40=−36 Since the discriminant is negative, the quadratic has no real roots and cannot be factored further using real coefficients.

Final Answer

x3−3*x2+12*x−10=(x−1)*(x2−2*x+10)

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