Certainly! Let's work through the problem step by step.
### Problem
We need to find the expression [tex]\(x^3 - y^3\)[/tex].
### Solution
The problem states that we need to simplify the expression [tex]\(x^3 - y^3\)[/tex].
### Step-by-Step Solution
1. Understanding the Expression:
- The expression [tex]\(x^3 - y^3\)[/tex] represents the difference of two cubes.
2. Difference of Cubes Formula:
- The formula for the difference of two cubes is:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
- Here, we can let [tex]\(a = x\)[/tex] and [tex]\(b = y\)[/tex].
3. Applying the Formula:
- Substitute [tex]\(a = x\)[/tex] and [tex]\(b = y\)[/tex] into the formula:
[tex]\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\][/tex]
4. Conclusion:
- Therefore, the expression [tex]\(x^3 - y^3\)[/tex] can be factored as:
[tex]\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\][/tex]
While we've shown the factorized form, the simplest form we started with is:
[tex]\[
x^3 - y^3
\][/tex]
So, the expression [tex]\(x^3 - y^3\)[/tex] is already in its simplest polynomial form.
