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

9-3 Multiplying Polynomials

In this Grade 9 Algebra 1 lesson from California Reveal Math, students learn to multiply polynomials using the vertical, horizontal, and box methods, applying the Distributive Property to produce quadratic expressions and higher-degree polynomial products. The lesson covers multiplying binomials and trinomials, combining like terms, and writing results in standard form. Real-world applications, such as calculating the area of a parking lot with a surrounding sidewalk, show students how polynomial multiplication models practical problems.

Section 1

Multiplying Binomials Using FOIL

Property

The FOIL method is a shortcut for multiplying two binomials. FOIL stands for First, Outer, Inner, Last — the four pairs of terms you multiply:

(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}

Section 2

Multiply a Binomial by a Binomial Using the Distributive Property

Property

To multiply $(x+3)(x+7)$ , you distribute the second binomial, $(x+7)$ , to each term of the first binomial.This gives $x(x+7) + 3(x+7)$ .
Then, you distribute again to get $x^2 + 7x + 3x + 21$ .
Finally, combine like terms to get $x^2 + 10x + 21$ .
Notice that you multiplied the two terms of the first binomial by the two terms of the second binomial, resulting in four multiplications.

Examples

To multiply $(a+4)(a+6)$ , distribute $(a+6)$ : $a(a+6) + 4(a+6) = a^2 + 6a + 4a + 24$ , which simplifies to $a^2 + 10a + 24$ .
For $(2x+1)(x-3)$ , distribute $(x-3)$ : $2x(x-3) + 1(x-3) = 2x^2 - 6x + x - 3$ , which simplifies to $2x^2 - 5x - 3$ .
To multiply $(y-5)(z+2)$ , distribute $(z+2)$ : $y(z+2) - 5(z+2) = yz + 2y - 5z - 10$ . There are no like terms to combine.

Explanation

This method breaks down the problem into smaller, familiar steps. You take the first term of the first binomial and multiply it by the entire second binomial, then do the same with the second term. It guarantees every piece gets multiplied.

Section 3

Multiply Binomials Using the Horizontal Method

Property

The horizontal method multiplies binomials by writing them side-by-side and systematically distributing each term from the first binomial to every term in the second binomial: $(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Multiplying Binomials Using FOIL

Property

The FOIL method is a shortcut for multiplying two binomials. FOIL stands for First, Outer, Inner, Last — the four pairs of terms you multiply:

(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}

Section 2

Multiply a Binomial by a Binomial Using the Distributive Property

Property

Examples

To multiply $(a+4)(a+6)$ , distribute $(a+6)$ : $a(a+6) + 4(a+6) = a^2 + 6a + 4a + 24$ , which simplifies to $a^2 + 10a + 24$ .
For $(2x+1)(x-3)$ , distribute $(x-3)$ : $2x(x-3) + 1(x-3) = 2x^2 - 6x + x - 3$ , which simplifies to $2x^2 - 5x - 3$ .
To multiply $(y-5)(z+2)$ , distribute $(z+2)$ : $y(z+2) - 5(z+2) = yz + 2y - 5z - 10$ . There are no like terms to combine.

Explanation

Section 3

Multiply Binomials Using the Horizontal Method

Property

Examples

Overview

Lesson overview

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

Overview

Section 1

Multiplying Binomials Using FOIL

Property

The FOIL method is a shortcut for multiplying two binomials. FOIL stands for First, Outer, Inner, Last — the four pairs of terms you multiply:

(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}

Section 2

Multiply a Binomial by a Binomial Using the Distributive Property

Property

Examples

To multiply $(a+4)(a+6)$ , distribute $(a+6)$ : $a(a+6) + 4(a+6) = a^2 + 6a + 4a + 24$ , which simplifies to $a^2 + 10a + 24$ .
For $(2x+1)(x-3)$ , distribute $(x-3)$ : $2x(x-3) + 1(x-3) = 2x^2 - 6x + x - 3$ , which simplifies to $2x^2 - 5x - 3$ .
To multiply $(y-5)(z+2)$ , distribute $(z+2)$ : $y(z+2) - 5(z+2) = yz + 2y - 5z - 10$ . There are no like terms to combine.