Find each product. (x+7y)(3x-5y)

Verified step by step guidance

1. Identify the problem as the product of two binomials: \((x + 7y)(3x - 5y)\).
2. Apply the distributive property (also known as FOIL method) to multiply each term in the first binomial by each term in the second binomial: First, Outer, Inner, Last.
3. Multiply the First terms: \(x \times 3x = 3x^{2}\).
4. Multiply the Outer terms: \(x \times (-5y) = -5xy\).
5. Multiply the Inner terms: \(7y \times 3x = 21xy\), and multiply the Last terms: \(7y \times (-5y) = -35y^{2}\). Then combine like terms \(-5xy\) and \$21xy$.

Key Concepts

Distributive Property

The distributive property allows you to multiply each term inside one parenthesis by each term inside the other. It is essential for expanding expressions like (x + 7y)(3x - 5y) by distributing each term in the first binomial across the second.

Combining Like Terms

After expanding the product, you combine like terms, which are terms with the same variables raised to the same powers. This simplifies the expression into its simplest form, making it easier to interpret or use in further calculations.

Multiplication of Variables and Coefficients

When multiplying terms like x and 3x or 7y and -5y, multiply the coefficients (numbers) and apply the laws of exponents to variables. For example, x * x = x², and coefficients multiply normally, which is crucial for correctly expanding the product.