In Exercises 9–22, multiply the monomial and the polynomial.3ab² (6a²b³+5ab)

Verified step by step guidance

1. Identify the expression to be multiplied: \(3ab^2\) and \((6a^2b^3 + 5ab)\).
2. Apply the distributive property: multiply \(3ab^2\) by each term inside the parentheses.
3. First, multiply \(3ab^2\) by \(6a^2b^3\): \(3ab^2 \times 6a^2b^3\).
4. Next, multiply \(3ab^2\) by \(5ab\): \(3ab^2 \times 5ab\).
5. Combine the results of the multiplications to form the expanded polynomial.

Key Concepts

Monomials

A monomial is a single term algebraic expression that consists of a coefficient and variables raised to non-negative integer powers. For example, in the expression 3ab², 3 is the coefficient, and ab² represents the variables a and b raised to the second power. Understanding monomials is essential for performing operations like multiplication with polynomials.

Polynomials

A polynomial is an algebraic expression that consists of one or more terms, where each term is a product of a coefficient and variables raised to non-negative integer powers. The expression 6a²b³ + 5ab is a polynomial with two terms. Recognizing the structure of polynomials is crucial for applying algebraic operations such as multiplication.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by each term in a polynomial. In the given problem, applying this property enables the multiplication of the monomial 3ab² with each term of the polynomial, resulting in a simplified expression. Mastery of this property is vital for efficiently solving algebraic expressions.