In Exercises 67–82, find each product. (3x−y)(2x+5y)

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Identify the problem as the multiplication of two binomials: $(3x - y)(2x + 5y)$. This requires the use of the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last).

Apply the FOIL method: Multiply the first terms of each binomial. This means multiplying $3x$ and $2x$, which gives $6x^2$.

Next, multiply the outer terms: $3x$ and $5y$, which gives $15xy$.

Then, multiply the inner terms: $-y$ and $2x$, which gives $-2xy$.

Finally, multiply the last terms: $-y$ and $5y$, which gives $-5y^2$. Combine all these terms to form the expanded expression: $6x^2 + 15xy - 2xy - 5y^2$. Simplify the middle terms to complete the solution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This principle allows us to multiply a single term by two or more terms inside parentheses. In the context of the given expression, it will be used to distribute each term in the first binomial across each term in the second binomial.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are algebraic expressions containing two terms. The result of multiplying two binomials is typically a polynomial with four terms, which can often be simplified. Understanding how to combine like terms after multiplication is crucial for arriving at the final answer.

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 using the distributive property to multiply the binomials, the resulting polynomial may contain like terms that can be combined to produce a more concise expression. This step is essential for presenting the final answer in its simplest form.

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