In Exercises 9–22, multiply the monomial and the polynomial.4xy(7x+3y)

Verified step by step guidance

1. First, distribute the monomial \(4xy\) to each term inside the parentheses.
2. Multiply \(4xy\) by the first term \(7x\) in the polynomial.
3. This results in \(4xy \cdot 7x = 28x^2y\).
4. Next, multiply \(4xy\) by the second term \(3y\) in the polynomial.
5. This results in \(4xy \cdot 3y = 12xy^2\).

Key Concepts

Monomials

A monomial is a single term algebraic expression that consists of a coefficient and one or more variables raised to non-negative integer powers. For example, in the expression 4xy, 4 is the coefficient, and x and y are the variables. Understanding monomials is essential for performing operations like multiplication with polynomials.

Polynomials

A polynomial is an algebraic expression that consists of multiple terms, which can include constants, variables, and non-negative integer exponents. The expression (7x + 3y) is a polynomial with two terms. Recognizing the structure of polynomials is crucial for applying algebraic operations such as distribution when multiplying with monomials.

Distribution

Distribution is a fundamental algebraic property that involves multiplying a single term by each term within a polynomial. This process is often referred to as the distributive property, expressed mathematically as a(b + c) = ab + ac. In the given problem, applying distribution allows us to multiply the monomial 4xy by each term in the polynomial (7x + 3y) to find the resulting expression.