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

Verified step by step guidance

1. Distribute the first term of the first binomial, \(3x\), to each term in the second binomial, \((2x + 5y)\).
2. Calculate \(3x \cdot 2x\) to get \(6x^2\).
3. Calculate \(3x \cdot 5y\) to get \(15xy\).
4. Distribute the second term of the first binomial, \(-y\), to each term in the second binomial, \((2x + 5y)\).
5. Calculate \(-y \cdot 2x\) to get \(-2xy\) and \(-y \cdot 5y\) to get \(-5y^2\).

Key Concepts

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 performing the multiplication of the binomials, it is essential to identify and combine any like terms to simplify the expression into its most concise form.