Lesson Objectives
• Learn how to find the product of two monomials
• Learn how to find the product of two binomials using FOIL
• Learn how to find the product of any two polynomials
• Learn how to find the product of three or more polynomials

## Multiplying 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.
(4x2)(-2x5)
We can reorder the multiplication:
(4 • -2)(x2 • x5)
-8x7
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(2x2 + 11)
We will distribute the 5x to each term inside of the parentheses.
5x • 2x2 + 5x • 11
(5 • 2)(x • x2) + (5 • 11)x
10x3 + 55x
When we multiply two non-monomials together, we also use the distributive 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
2x2 + x + 16x + 8
2x2 + 17x + 8
Example 4: Find each product.
(8x + 5y)(5x2 - 3xy + 3y2)
We will find the sum of each term of the first polynomial multiplied by each term of the second polynomial:
8x(5x2 - 3xy + 3y2) + 5y(5x2 - 3xy + 3y2)
8x • 5x2 + 8x • -3xy + 8x • 3y2 + 5y • 5x2 + 5y • -3xy + 5y • 3y2
40x3 - 24x2y + 24xy2 + 25x2y - 15xy2 + 15y3
40x3 + x2y + 9xy2 + 15y3

### 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 = 30x2
O » 5x • 5y = 25xy
I » 7y • 6x = 42xy
L » 7y • 5y = 35y2
We will find the sum of these individual products.
30x2 + 25xy + 42xy + 35y2
30x2 + 67xy + 35y2

### 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)(5x2 + 9)
Let's begin by finding the product of the first two (leftmost) polynomials:
(2x + 5)(6x - 8)
12x2 - 16x + 30x - 40
12x2 + 14x - 40
Now we can multiply the result by the last polynomial:
(12x2 + 14x - 40)(5x2 + 9)
5x2(12x2 + 14x - 40) + 9(12x2 + 14x - 40)
60x4 + 70x3 - 200x2 + 108x2 + 126x - 360
60x4 + 70x3 - 92x2 + 126x - 360

#### Skills Check:

Example #1

Find each product $$3(x+2)$$

A
$$16x^2 + 16x$$
B
$$9x^4 + 6x^3$$
C
$$6x^2 - 12x$$
D
$$3x + 6$$
E
$$6x^2 + x$$

Example #2

Find each product $$(4x+6)(2x+5)$$

A
$$8x^2 + 32x + 30$$
B
$$-15x^2 + 18x - 3$$
C
$$8x^2 + 30$$
D
$$8x^2 + 48x + 3$$
E
$$12x^2 + 5x + 1$$

Example #3

Find each product $$(x - 3)(7x^2 - 5x - 2)$$

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A
$$6x^3 - 17x^2 - 20x - 32$$
B
$$12x^3 - 52x^2 + 5x + 25$$
C
$$7x^3 + 36x^2 + 2x - 15$$
D
$$7x^3 - 26x^2 + 13x + 6$$
E
$$-3x^3 + 14x^2 - 5x - 3$$

Example #4

Find each product $$5x^2(3x - 1)(2x^2 - 7x + 4)$$

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A
$$5x^6 - x^2 - 10$$
B
$$25x^5 + 19x^4 + 8x^3 - 11x^2$$
C
$$30x^5 - 115x^4 + 95x^3 - 20x^2$$
D
$$5x^3 + 10x^2 - 5x - 1$$
E
$$20x^6 + 121x^3 - 14x + 115$$           