In Exercises 67–82, find each product. (x−y)(x²+xy+y²)

Verified step by step guidance

1. Step 1: Recognize that the problem involves multiplying two expressions: \((x - y)\) and \((x^2 + xy + y^2)\). This is a straightforward application of the distributive property.
2. Step 2: Apply the distributive property by multiplying \((x - y)\) with each term in \((x^2 + xy + y^2)\). This means you will distribute \(x\) and \(-y\) across \((x^2 + xy + y^2)\).
3. Step 3: First, distribute \(x\) to each term in \((x^2 + xy + y^2)\): \(x \cdot x^2 + x \cdot xy + x \cdot y^2\), which simplifies to \(x^3 + x^2y + xy^2\).
4. Step 4: Next, distribute \(-y\) to each term in \((x^2 + xy + y^2)\): \(-y \cdot x^2 - y \cdot xy - y \cdot y^2\), which simplifies to \(-x^2y - xy^2 - y^3\).
5. Step 5: Combine all the terms from Step 3 and Step 4: \(x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3\). Then, simplify by combining like terms to get the final expression.

Key Concepts

Factoring and Distributing

Factoring involves breaking down expressions into simpler components, while distributing refers to applying the distributive property to multiply terms. In this case, we need to distribute (x - y) across the polynomial (x^2 + xy + y^2) to find the product. Understanding how to apply these operations is crucial for simplifying algebraic expressions.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together, which requires combining like terms and applying the distributive property. Each term in the first polynomial must be multiplied by each term in the second polynomial. This concept is essential for solving the given expression, as it allows us to systematically find the resulting polynomial.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. After distributing and multiplying the polynomials, it is important to identify and combine these like terms to arrive at the final simplified expression. This step is key to ensuring the answer is presented in its simplest form.