Lesson Objectives

- Demonstrate an understanding of the commutative property of multiplication
- Demonstrate an understanding of the associative property of multiplication
- Learn how to find the product of two polynomials
- Learn how to find the product of two binomials using FOIL
- Learn how to find the product of more than two polynomials

## How to Multiply Polynomials

In our last lesson, we gave a basic definition of a polynomial, and we learned how to
add and subtract polynomials.
In this lesson, we will focus on how to multiply polynomials. We can multiply two monomials together using the
associative and commutative properties of multiplication. Let's look at an example.

Example 1: Find each product

(4x

We can reorder the multiplication:

(4 • -2)(x

-8x

When we multiply a monomial by a polynomial that is not a monomial, we use the distributive property. Let's look at an example.

Example 2: Find each product

5x(2x

We will distribute the 5x to each term inside of the parentheses.

5x • 2x

(5 • 2)(x • x

10x

When we multiply two non-monomials together, we also use the distribute property. We will form the sum of each term of the first polynomial multiplied by each term of the second polynomial. Let's look at an example.

Example 3: Find each product

(x + 8)(2x + 1)

We will find the sum of each term of the first polynomial multiplied by each term of the second polynomial:

x(2x + 1) + 8(2x + 1)

x • 2x + x • 1 + 8 • 2x + 8 • 1

2x

2x

Example 4: Find each product

(8x + 5y)(5x

We will find the sum of each term of the first polynomial multiplied by each term of the second polynomial:

8x(5x

8x • 5x

40x

40x

F » First Terms

O » Outer Terms

I » Inner Terms

L » Last Terms

To use the FOIL technique, we find the sum of the first terms, outer terms, inner terms, and last terms. Let's look at an example.

Example 5: Find each product using FOIL

(5x + 7y)(6x + 5y)

F » 5x • 6x = 30x

O » 5x • 5y = 25xy

I » 7y • 6x = 42xy

L » 7y • 5y = 35y

We will find the sum of these individual products.

30x

30x

Example 6: Find each product

(2x + 5)(6x - 8)(5x

Let's begin by finding the product of the first two (leftmost) polynomials:

(2x + 5)(6x - 8)

12x

12x

Now we can multiply the result by the last polynomial:

(12x

5x

60x

60x

Example 1: Find each product

(4x

^{2})(-2x^{5})We can reorder the multiplication:

(4 • -2)(x

^{2}• x^{5})-8x

^{7}When we multiply a monomial by a polynomial that is not a monomial, we use the distributive property. Let's look at an example.

Example 2: Find each product

5x(2x

^{2}+ 11)We will distribute the 5x to each term inside of the parentheses.

5x • 2x

^{2}+ 5x • 11(5 • 2)(x • x

^{2}) + (5 • 11)x10x

^{3}+ 55xWhen we multiply two non-monomials together, we also use the distribute property. We will form the sum of each term of the first polynomial multiplied by each term of the second polynomial. Let's look at an example.

Example 3: Find each product

(x + 8)(2x + 1)

We will find the sum of each term of the first polynomial multiplied by each term of the second polynomial:

x(2x + 1) + 8(2x + 1)

x • 2x + x • 1 + 8 • 2x + 8 • 1

2x

^{2}+ x + 16x + 82x

^{2}+ 17x + 8Example 4: Find each product

(8x + 5y)(5x

^{2}- 3xy + 3y^{2})We will find the sum of each term of the first polynomial multiplied by each term of the second polynomial:

8x(5x

^{2}- 3xy + 3y^{2}) + 5y(5x^{2}- 3xy + 3y^{2})8x • 5x

^{2}+ 8x • -3xy + 8x • 3y^{2}+ 5y • 5x^{2}+ 5y • -3xy + 5y • 3y^{2}40x

^{3}- 24x^{2}y + 24xy^{2}+ 25x^{2}y - 15xy^{2}+ 15y^{2}40x

^{3}+ x^{2}y + 9xy^{2}+ 15y^{3}### Multiplying two Binomials using FOIL

We will often have to find the product of two binomials. When this situation occurs, we can use the FOIL technique.F » First Terms

O » Outer Terms

I » Inner Terms

L » Last Terms

To use the FOIL technique, we find the sum of the first terms, outer terms, inner terms, and last terms. Let's look at an example.

Example 5: Find each product using FOIL

(5x + 7y)(6x + 5y)

F » 5x • 6x = 30x

^{2}O » 5x • 5y = 25xy

I » 7y • 6x = 42xy

L » 7y • 5y = 35y

^{2}We will find the sum of these individual products.

30x

^{2}+ 25xy + 42xy + 35y^{2}30x

^{2}+ 67xy + 35y^{2}### Multiplying More Than Two Polynomials

We can find the product of more than two polynomials by multiplying pairs of polynomials until we have our product. Let's look at an example.Example 6: Find each product

(2x + 5)(6x - 8)(5x

^{2}+ 9)Let's begin by finding the product of the first two (leftmost) polynomials:

(2x + 5)(6x - 8)

12x

^{2}- 16x + 30x - 4012x

^{2}+ 14x - 40Now we can multiply the result by the last polynomial:

(12x

^{2}+ 14x - 40)(5x^{2}+ 9)5x

^{2}(12x^{2}+ 14x - 40) + 9(12x^{2}+ 14x - 40)60x

^{4}+ 70x^{3}- 200x^{2}+ 108x^{2}+ 126x - 36060x

^{4}+ 70x^{3}- 92x^{2}+ 126x - 360
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