Factor x^8-1

Problem

x^8−1

Solution

1. Identify the expression as a difference of squares, which follows the pattern a^2−b^2=(a−b)*(a+b)

2. Apply the formula to the initial expression by treating x^8 as (x^4)^2 and 1 as 1^2

x^8−1=(x^4−1)*(x^4+1)

3. Factor the resulting term (x^4−1) as another difference of squares.

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

4. Factor the term (x^2−1) as a difference of squares one last time.

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

Final Answer

x^8−1=(x−1)*(x+1)*(x^2+1)*(x^4+1)

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