Factor 2x^4-13x^3+6x^2+64x-32

Problem

2*x4−13*x3+6*x2+64*x−32

Solution

1. Identify possible rational roots using the Rational Root Theorem, testing factors of the constant term −32 divided by factors of the leading coefficient 2

2. Test the value x=4 using synthetic division or direct substitution.

2*(4)4−13*(4)3+6*(4)2+64*(4)−32=512−832+96+256−32=0

3. Divide the polynomial by (x−4) to obtain the depressed polynomial.

(2*x4−13*x3+6*x2+64*x−32)÷(x−4)=2*x3−5*x2−14*x+8

4. Test the value x=−2 in the new polynomial 2*x3−5*x2−14*x+8

2*(−2)3−5*(−2)2−14*(−2)+8=−16−20+28+8=0

5. Divide the cubic polynomial by (x+2) to find the remaining quadratic factor.

(2*x3−5*x2−14*x+8)÷(x+2)=2*x2−9*x+4

6. Factor the quadratic expression 2*x2−9*x+4 by splitting the middle term or using the quadratic formula.

2*x2−8*x−x+4=2*x*(x−4)−1*(x−4)=(2*x−1)*(x−4)

7. Combine all the factors found in the previous steps.

(x−4)*(x+2)*(2*x−1)*(x−4)=(x−4)2*(x+2)*(2*x−1)

Final Answer

2*x4−13*x3+6*x2+64*x−32=(x−4)2*(x+2)*(2*x−1)

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