Learn on PengiCalifornia Reveal Math, Algebra 1Unit 9: Polynomials

9-2 Multiplying Polynomials by Monomials

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to multiply polynomials by monomials using the Distributive Property to simplify expressions such as -2x(4x² + 3x - 5). The lesson covers distributing monomials with negative coefficients, combining like terms, and applying the technique to solve equations and real-world area problems involving trapezoids and rectangular prisms. Students also explore why the set of polynomials is closed under multiplication.

Section 1

Multiplying Monomials

Property

To multiply two monomials, rearrange the factors to group together the numerical coefficients and the powers of each base.
Then, multiply the coefficients and use the first law of exponents for the variable factors.

Examples

To multiply $(3a^2b)(4a^3b^4)$ , we group and multiply: $(3 \cdot 4)(a^2 \cdot a^3)(b \cdot b^4) = 12a^5b^5$ .

The product of $(-6x^3y^2)(2x^5y)$ is found by multiplying coefficients and adding exponents of like bases: $(-6 \cdot 2)(x^3 \cdot x^5)(y^2 \cdot y) = -12x^8y^3$ .

Section 2

Multiplying by a monomial

Property

To multiply a polynomial by a monomial, we use the distributive law. This means we multiply each term of the polynomial by the monomial.

a(b + c) = ab + ac

Examples

To multiply $4x(2x^2 - 5x + 3)$ , distribute $4x$ to each term: $4x(2x^2) + 4x(-5x) + 4x(3) = 8x^3 - 20x^2 + 12x$ .

Section 3

Distributing a Negative Monomial Over a Polynomial

Property

When multiplying a polynomial by a negative monomial, apply the distributive property to every term in the polynomial, including the negative sign:

-a(b + c) = -ab + (-ac) = -ab - ac

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Multiplying Monomials

Property

Examples

To multiply $(3a^2b)(4a^3b^4)$ , we group and multiply: $(3 \cdot 4)(a^2 \cdot a^3)(b \cdot b^4) = 12a^5b^5$ .

The product of $(-6x^3y^2)(2x^5y)$ is found by multiplying coefficients and adding exponents of like bases: $(-6 \cdot 2)(x^3 \cdot x^5)(y^2 \cdot y) = -12x^8y^3$ .

Section 2

Multiplying by a monomial

Property

To multiply a polynomial by a monomial, we use the distributive law. This means we multiply each term of the polynomial by the monomial.

a(b + c) = ab + ac

Examples

To multiply $4x(2x^2 - 5x + 3)$ , distribute $4x$ to each term: $4x(2x^2) + 4x(-5x) + 4x(3) = 8x^3 - 20x^2 + 12x$ .

Section 3

Distributing a Negative Monomial Over a Polynomial

Property

When multiplying a polynomial by a negative monomial, apply the distributive property to every term in the polynomial, including the negative sign:

-a(b + c) = -ab + (-ac) = -ab - ac

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Multiplying Monomials

Property

Examples

To multiply $(3a^2b)(4a^3b^4)$ , we group and multiply: $(3 \cdot 4)(a^2 \cdot a^3)(b \cdot b^4) = 12a^5b^5$ .

The product of $(-6x^3y^2)(2x^5y)$ is found by multiplying coefficients and adding exponents of like bases: $(-6 \cdot 2)(x^3 \cdot x^5)(y^2 \cdot y) = -12x^8y^3$ .

Section 2

Multiplying by a monomial

Property

To multiply a polynomial by a monomial, we use the distributive law. This means we multiply each term of the polynomial by the monomial.

a(b + c) = ab + ac

Examples

To multiply $4x(2x^2 - 5x + 3)$ , distribute $4x$ to each term: $4x(2x^2) + 4x(-5x) + 4x(3) = 8x^3 - 20x^2 + 12x$ .

Section 3

Distributing a Negative Monomial Over a Polynomial

Property

When multiplying a polynomial by a negative monomial, apply the distributive property to every term in the polynomial, including the negative sign:

-a(b + c) = -ab + (-ac) = -ab - ac