Factor x^4-2x^3-8x+16

Problem

x^4−2*x^3−8*x+16

Solution

1. Group the terms into two pairs to prepare for factoring by grouping.

(x^4−2*x^3)+(−8*x+16)

2. Factor out the greatest common factor from each pair.

x^3*(x−2)−8*(x−2)

3. Factor out the common binomial (x−2) from the expression.

(x^3−8)*(x−2)

4. Identify the difference of cubes in the first factor, noting that 8=2^3

(x^3−2^3)*(x−2)

5. Apply the difference of cubes formula a^3−b^3=(a−b)*(a^2+a*b+b^2) to factor x^3−2^3

(x−2)*(x^2+2*x+4)*(x−2)

6. Combine the identical binomial factors to simplify the final expression.

(x−2)^2*(x^2+2*x+4)

Final Answer

x^4−2*x^3−8*x+16=(x−2)^2*(x^2+2*x+4)

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