In Exercises 9–22, multiply the monomial and the polynomial. 4x²(3x+2)

Verified step by step guidance

1. Identify the monomial and the polynomial: The monomial is \(4x^2\) and the polynomial is \(3x + 2\).
2. Apply the distributive property: Multiply the monomial \(4x^2\) by each term in the polynomial \(3x + 2\).
3. First, multiply \(4x^2\) by \(3x\): Use the rule \(a^m \cdot a^n = a^{m+n}\) to combine the exponents of \(x\).
4. Next, multiply \(4x^2\) by \(2\): Simply multiply the coefficients and keep the \(x^2\) term as it is.
5. Combine the results from the previous steps to express the final expanded polynomial.

Key Concepts

Monomials

A monomial is a mathematical expression that consists of a single term, which can be a number, a variable, or a product of numbers and variables raised to non-negative integer powers. In the given question, '4x²' is a monomial, representing a coefficient (4) multiplied by a variable (x) raised to the power of 2.

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 '3x + 2' is a polynomial with two terms, known as a binomial, which can be added or subtracted to form more complex expressions.

Distributive Property

The distributive property is a fundamental algebraic principle that states a(b + c) = ab + ac. This property allows us to multiply a monomial by each term in a polynomial. In the exercise, applying the distributive property means multiplying '4x²' by both '3x' and '2' to find the resulting expression.