Factor x^6-y^6

Problem

x^6−y^6

Solution

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

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

x^6−y^6=(x^3−y^3)*(x^3+y^3)

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

x^3−y^3=(x−y)*(x^2+x*y+y^2)

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

x^3+y^3=(x+y)*(x^2−x*y+y^2)

5. Combine all the factors into a single expression.

(x−y)*(x^2+x*y+y^2)*(x+y)*(x^2−x*y+y^2)

Final Answer

x^6−y^6=(x−y)*(x+y)*(x^2+x*y+y^2)*(x^2−x*y+y^2)

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