Factor x⁶ - y⁶ as difference of two squares

To factor \(x^6 - y^6\) as a difference of two squares, we can apply the difference of squares formula multiple times. The difference of squares formula states:

\[
a^2 - b^2 = (a - b)(a + b)
\]

### Step-by-Step Solution:

1. **Recognize the Expression as a Difference of Squares:**

   \[
   x^6 - y^6 = (x^3)^2 - (y^3)^2
   \]

   Here, \(a = x^3\) and \(b = y^3\).

2. **Apply the Difference of Squares Formula:**

   \[
   (x^3)^2 - (y^3)^2 = (x^3 - y^3)(x^3 + y^3)
   \]

3. **Factor Each Term Further:**

   Now, we need to factor \(x^3 - y^3\) and \(x^3 + y^3\).

   - **Factor \(x^3 - y^3\):** This is a difference of cubes, and it factors as:
   
   \[
   x^3 - y^3 = (x - y)(x^2 + xy + y^2)
   \]

   - **Factor \(x^3 + y^3\):** This is a sum of cubes, and it factors as:

   \[
   x^3 + y^3 = (x + y)(x^2 - xy + y^2)
   \]

4. **Combine All Factors:**

   Substitute the factored forms back into the expression:

   \[
   x^6 - y^6 = (x^3 - y^3)(x^3 + y^3) = (x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2)
   \]

### Final Factored Form:

Thus, the fully factored form of \(x^6 - y^6\) is:

\[
x^6 - y^6 = (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)
\]

Would you like a further breakdown of the steps or have any other questions?

### Further Questions to Explore:

1. How would you factor \(x^4 - y^4\) using the difference of squares?
2. Can you factor \(x^8 - y^8\) using a similar method?
3. How would you apply the difference of squares formula to \(x^2 - 1\)?
4. What is the factored form of \(x^6 + y^6\)?
5. How does the factoring process change for expressions like \(x^6 - z^6\)?

### Tip:

When factoring higher powers, repeatedly apply the difference of squares and other algebraic identities, such as the sum and difference of cubes, to break down the expression step by step.

Learn how to factor the polynomial x⁶ - y⁶ as a difference of two squares with a detailed step-by-step solution. This problem explores key algebraic concepts such as factoring, polynomials, and the difference of squares. Suitable for high school students in Grades 9-12.

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