Lesson Objectives

- Demonstrate an understanding of polynomials
- Demonstrate an understanding of the rules of exponents
- Learn how to multiply a monomial by a polynomial
- Learn how to multiply two polynomials together
- Learn how to multiply more than two polynomials together

## How to Multiply Polynomials

When we multiply two or more polynomials together, we use the distributive property and our product rule for exponents. Recall the product rule for
exponents tells us to keep the base the same and add exponents when we multiply two expressions in exponent form that have the same base.
Recall the distributive property allows us to
distribute multiplication over addition or subtraction. If we start with a simple case, such as a monomial times a polynomial, we will
multiply the monomial by each term of the polynomial using the distributive property. Let's look at an example.

Example 1: Find each product

9(8x + 2)

Use the distributive property and multiply 9 by each term inside of the parentheses:

9(8x + 2) = 9 • 8x + 9 • 2 = 72x + 18

Example 2: Find each product

-3x

Use the distributive property and multiply (-3x

-3x

When we multiply two polynomials together and neither is a monomial, we multiply each term of the first polynomial by each term of the second polynomial. Once this is done, we add our products (combine like terms) to arrive at our answer. Let's take a look at an example.

Example 3: Find each product

(5x - 14)(8x - 8)

We will multiply each term of the first polynomial (5x and (-14)) by each term of the second polynomial. We will then add the products:

5x(8x - 8) + (-14)(8x - 8) =

5x • 8x + 5x • (-8) + (-14) • 8x + (-14) • (-8) =

40x

Combine Like Terms:

40x

Example 4: Find each product

(7x - 1)(4x

We will multiply each term of the first polynomial (7x and (-1)) by each term of the second polynomial. We will then add the products:

7x(4x

7x • 4x

28x

Combine Like Terms:

28x

5 • 3 • 2

We can multiply any two numbers together first and then multiply that product by the final number:

5 • 3 • 2 = 15 • 2 = 30

5 • 3 • 2 = 5 • 6 = 30

We can use the same technique when multiplying three or more polynomials. Let's take a look at an example.

Example 5: Find each product

(2x

Let's focus on one multiplication at a time. We will begin with the multiplication of the two leftmost polynomials:

(2x

2x

6x

Combine Like Terms:

6x

Now we multiply this result by the third polynomial:

(-9x

-9x

-54x

Combine Like Terms:

-54x

Example 1: Find each product

9(8x + 2)

Use the distributive property and multiply 9 by each term inside of the parentheses:

9(8x + 2) = 9 • 8x + 9 • 2 = 72x + 18

Example 2: Find each product

-3x

^{2}(2x - 7)Use the distributive property and multiply (-3x

^{2}) by each term inside of the parentheses:-3x

^{2}• 2x + (-3x^{2}) • (-7) = -6x^{3}+ 21x^{2}When we multiply two polynomials together and neither is a monomial, we multiply each term of the first polynomial by each term of the second polynomial. Once this is done, we add our products (combine like terms) to arrive at our answer. Let's take a look at an example.

Example 3: Find each product

(5x - 14)(8x - 8)

We will multiply each term of the first polynomial (5x and (-14)) by each term of the second polynomial. We will then add the products:

5x(8x - 8) + (-14)(8x - 8) =

5x • 8x + 5x • (-8) + (-14) • 8x + (-14) • (-8) =

40x

^{2}- 40x - 112x + 112Combine Like Terms:

40x

^{2}- 152x + 112Example 4: Find each product

(7x - 1)(4x

^{2}- 2x + 9)We will multiply each term of the first polynomial (7x and (-1)) by each term of the second polynomial. We will then add the products:

7x(4x

^{2}- 2x + 9) + (-1)(4x^{2}- 2x + 9) =7x • 4x

^{2}+ 7x • (-2x) + 7x • 9 + (-1) • 4x^{2}+ (-1) • (-2x) + (-1) • 928x

^{3}- 14x^{2}+ 63x - 4x^{2}+ 2x - 9Combine Like Terms:

28x

^{3}- 18x^{2}+ 65x - 9### Multiplying More than Two Polynomials Together

What happens when we want to multiply three or more polynomials together? Think about multiplying more than two numbers together:5 • 3 • 2

We can multiply any two numbers together first and then multiply that product by the final number:

5 • 3 • 2 = 15 • 2 = 30

5 • 3 • 2 = 5 • 6 = 30

We can use the same technique when multiplying three or more polynomials. Let's take a look at an example.

Example 5: Find each product

(2x

^{2}- 5)(3x^{2}- 1)(-9x^{2}+ 8)Let's focus on one multiplication at a time. We will begin with the multiplication of the two leftmost polynomials:

(2x

^{2}- 5)(3x^{2}- 1) =2x

^{2}(3x^{2}- 1) + (-5)(3x^{2}- 1) =6x

^{4}- 2x^{2}+ (-15)x^{2}+ 5Combine Like Terms:

6x

^{4}- 17x^{2}+ 5Now we multiply this result by the third polynomial:

(-9x

^{2}+ 8)(6x^{4}- 17x^{2}+ 5) =-9x

^{2}(6x^{4}- 17x^{2}+ 5) + 8(6x^{4}- 17x^{2}+ 5) =-54x

^{6}+ 153x^{4}- 45x^{2}+ 48x^{4}- 136x^{2}+ 40Combine Like Terms:

-54x

^{6}+ 201x^{4}- 181x^{2}+ 40
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