To multiply \((x - y)(x^2 + 2xy + y^2)\), we will use the distributive property (also known as the FOIL method for binomials).

1. Distribute \(x\) across the second polynomial:
   - \(x \cdot x^2 = x^3\)
   - \(x \cdot 2xy = 2x^2y\)
   - \(x \cdot y^2 = xy^2\)

2. Distribute \(-y\) across the second polynomial:
   - \(-y \cdot x^2 = -yx^2\)
   - \(-y \cdot 2xy = -2xy^2\)
   - \(-y \cdot y^2 = -y^3\)

Now, combine all the terms:

\[
x^3 + 2x^2y + xy^2 - yx^2 - 2xy^2 - y^3
\]

Next, combine like terms:

- \(2x^2y - yx^2 = x^2y\) (since \(yx^2\) is the same as \(x^2y\))
- The other terms are already simplified.

Thus, the final result is:

\[
x^3 + x^2y - xy^2 - y^3
\]

If you have any questions or want more details, feel free to ask! Here are some related questions:

1. What would the result be if we used \( (x+y)(x^2-2xy+y^2) \)?
2. Can you show how to factor \( x^3 + x^2y - xy^2 - y^3 \)?
3. How can we verify the result using substitution?
4. What is the geometric interpretation of this polynomial?
5. How would you multiply \((a - b)(a^2 + ab + b^2)\) and what would the result be?

**Tip:** Always check for like terms when combining after distribution to simplify your polynomial efficiently.

Discover how to multiply the binomials (x-y)(x²+2xy+y²) using the distributive property. This solution provides a detailed step-by-step breakdown, simplifying to x³ + x²y - xy² - y³, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation