Factor x^4-2x^3+13x^2-32x-48

Problem

x^4−2*x^3+13*x^2−32*x−48

Solution

1. Identify potential rational roots using the Rational Root Theorem, testing factors of the constant term −48

2. Test x=−1 by substituting it into the polynomial: (−1)^4−2*(−1)^3+13*(−1)^2−32*(−1)−48=1+2+13+32−48=0

3. Divide the polynomial by (x+1) using synthetic division to obtain the quotient x^3−3*x^2+16*x−48

4. Factor the resulting cubic polynomial x^3−3*x^2+16*x−48 by grouping terms.

5. Group the terms as (x^3−3*x^2)+(16*x−48)

6. Factor out the greatest common factor from each group: x^2*(x−3)+16*(x−3)

7. Extract the common binomial factor (x−3) to get (x−3)*(x^2+16)

8. Combine all factors to write the completely factored form over the real numbers.

Final Answer

x^4−2*x^3+13*x^2−32*x−48=(x+1)*(x−3)*(x^2+16)

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