In Exercises 67–82, find each product. (x−y)(x^2+xy+y^2)

Verified step by step guidance

1. Identify the expression to be expanded: $(x-y)(x^2+xy+y^2)$.
2. Apply the distributive property (also known as the FOIL method for binomials) to multiply $(x-y)$ by each term in $(x^2+xy+y^2)$.
3. First, multiply $x$ by each term in $(x^2+xy+y^2)$: $x\cdot x^2$, $x\cdot xy$, and $x\cdot y^2$.
4. Next, multiply $-y$ by each term in $(x^2+xy+y^2)$: $-y\cdot x^2$, $-y\cdot xy$, and $-y\cdot y^2$.
5. Combine all the terms obtained from the multiplication, and simplify by combining like terms if necessary.

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 can result in a new polynomial. Each term in the first polynomial must be multiplied by each term in the second polynomial, and like terms are then combined. This concept is essential for solving the given problem, as it requires careful multiplication of the terms.

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 any like terms to arrive at the final simplified expression. This step is key to presenting the product in its simplest form.