Factor x^6-1

Problem

x6−1

Solution

1. Identify the expression as a difference of squares, since x6=(x3)2 and 1=1

2. Apply the formula for the difference of squares, a2−b2=(a−b)*(a+b) where a=x3 and b=1

x6−1=(x3−1)*(x3+1)

3. Factor the first term (x3−1) using the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2)

x3−1=(x−1)*(x2+x+1)

4. Factor the second term (x3+1) using the sum of cubes formula, a3+b3=(a+b)*(a2−a*b+b2)

x3+1=(x+1)*(x2−x+1)

5. Combine all the factors into a single expression.

(x−1)*(x2+x+1)*(x+1)*(x2−x+1)

Final Answer

x6−1=(x−1)*(x+1)*(x2+x+1)*(x2−x+1)

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