Factor x^6-1

Problem

x^6−1

Solution

1. Identify the expression as a difference of squares, since x^6=(x^3)^2 and 1=1^2

2. Apply the formula for the difference of squares, a^2−b^2=(a−b)*(a+b) where a=x^3 and b=1

x^6−1=(x^3−1)*(x^3+1)

3. Factor the first term (x^3−1) using the difference of cubes formula, a^3−b^3=(a−b)*(a^2+a*b+b^2)

x^3−1=(x−1)*(x^2+x+1)

4. Factor the second term (x^3+1) using the sum of cubes formula, a^3+b^3=(a+b)*(a^2−a*b+b^2)

x^3+1=(x+1)*(x^2−x+1)

5. Combine all the factors into a single expression.

(x−1)*(x^2+x+1)*(x+1)*(x^2−x+1)

Final Answer

x^6−1=(x−1)*(x+1)*(x^2+x+1)*(x^2−x+1)

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