Lesson Objectives

- Demonstrate an understanding of how to multiply polynomials
- Learn how to multiply two binomials together using FOIL
- Learn how to multiply more than two binomials together using FOIL

## How to Multiply Two Binomials Together using FOIL

In our last lesson, we learned how to multiply polynomials. In this lesson, we will learn how to find the product of two binomials using FOIL. The acronym FOIL stands for:

F » First Terms

O » Outer Terms

I » Inner Terms

L » Last Terms

Essentially, we will multiply the first terms together, the outer terms, the inner terms, and the last terms. Once we have found these four products, we will combine any like terms and report our answer. Let's look at a few examples.

Example 1: Find each product using the FOIL method.

(2x - 7)(6x + 6)

Using the FOIL method:

F » 2x • 6x = 12x

O » 2x • 6 = 12x

I » -7 • 6x = -42x

L » -7 • 6 = -42

Now we can write our four products together and combine any like terms:

12x

12x

Example 2: Find each product using the FOIL method.

(6x - 1)(7x + 3)

Using the FOIL method:

F » 6x • 7x = 42x

O » 6x • 3 = 18x

I » -1 • 7x = -7x

L » -1 • 3 = -3

Now we can write our four products together and combine any like terms:

42x

42x

Example 3: Find each product using the FOIL method.

(x + 3)(4x - 7)

Using the FOIL method:

F » x • 4x = 4x

O » x • -7 = -7x

I » 3 • 4x = 12x

L » 3 • -7 = -21

Now we can write our four products together and combine any like terms:

4x

4x

Example 4: Find each product.

3x

We can use FOIL for the two binomials at the end:

(7x - 5)(2x + 11) =

14x

14x

Now we can multiply 3x

3x

42x

Example 5: Find each product.

(2x - 1)(3x + 7)(9x - 5)

We can use FOIL to find the product of any two binomial factors. Let's use FOIL to find the product of the first two factors.

(2x - 1)(3x + 7) =

6x

6x

Now we can multiply our result by our third binomial:

(9x - 5)(6x

9x(6x

54x

54x

F » First Terms

O » Outer Terms

I » Inner Terms

L » Last Terms

Essentially, we will multiply the first terms together, the outer terms, the inner terms, and the last terms. Once we have found these four products, we will combine any like terms and report our answer. Let's look at a few examples.

Example 1: Find each product using the FOIL method.

(2x - 7)(6x + 6)

Using the FOIL method:

F » 2x • 6x = 12x

^{2}O » 2x • 6 = 12x

I » -7 • 6x = -42x

L » -7 • 6 = -42

Now we can write our four products together and combine any like terms:

12x

^{2}+ 12x - 42x - 42 =12x

^{2}- 30x - 42Example 2: Find each product using the FOIL method.

(6x - 1)(7x + 3)

Using the FOIL method:

F » 6x • 7x = 42x

^{2}O » 6x • 3 = 18x

I » -1 • 7x = -7x

L » -1 • 3 = -3

Now we can write our four products together and combine any like terms:

42x

^{2}+ 18x - 7x - 3 =42x

^{2}+ 11x - 3Example 3: Find each product using the FOIL method.

(x + 3)(4x - 7)

Using the FOIL method:

F » x • 4x = 4x

^{2}O » x • -7 = -7x

I » 3 • 4x = 12x

L » 3 • -7 = -21

Now we can write our four products together and combine any like terms:

4x

^{2}- 7x + 12x - 21 =4x

^{2}+ 5x - 21### Multiplying More than Two Binomials using FOIL

Unfortunately, we can only use FOIL for the product of two binomials. The formula does not work when we apply it to other multiplication problems with polynomials. If we encounter a multiplication problem with at least two binomials, we can use FOIL to multiply two binomials together and then multiply that result by the remaining factors using the distributive property. Let's look at a few examples.Example 4: Find each product.

3x

^{2}(7x - 5)(2x + 11)We can use FOIL for the two binomials at the end:

(7x - 5)(2x + 11) =

14x

^{2}+ 77x - 10x - 55 =14x

^{2}+ 67x - 55Now we can multiply 3x

^{2}by the result:3x

^{2}(14x^{2}+ 67x - 55) =42x

^{4}+ 201x^{3}- 165x^{2}Example 5: Find each product.

(2x - 1)(3x + 7)(9x - 5)

We can use FOIL to find the product of any two binomial factors. Let's use FOIL to find the product of the first two factors.

(2x - 1)(3x + 7) =

6x

^{2}+ 14x - 3x - 7 =6x

^{2}+ 11x - 7Now we can multiply our result by our third binomial:

(9x - 5)(6x

^{2}+ 11x - 7) =9x(6x

^{2}+ 11x - 7) + (-5)(6x^{2}+ 11x - 7) =54x

^{3}+ 99x^{2}- 63x - 30x^{2}- 55x + 35 =54x

^{3}+ 69x^{2}- 118x + 35#### Skills Check:

Example #1

Find each product. $$(4x - 5)(8x - 1)$$

Please choose the best answer.

A

$$9x^2 - 5x + 4$$

B

$$30x^2 - 2x + 1$$

C

$$32x^2 - 44x + 5$$

D

$$32x^2 - 36x - 5$$

E

$$4x^2 - 6x - 3$$

Example #2

Find each product. $$(2x + 3)(x - 5)$$

Please choose the best answer.

A

$$2x^{2}- 7x - 15$$

B

$$42x^{2}- 12$$

C

$$7x^{2}+ 30x - 25$$

D

$$42x^{2}- 10x - 12$$

E

$$-2x^{2}- 15x + 15$$

Example #3

Find each product. $$(x - y)(x - 5y)$$

Please choose the best answer.

A

$$2x^{2}+ 2xy - 24y^{2}$$

B

$$x^{2}- 6xy + 5y^{2}$$

C

$$x^{2}+ 5y^{2}$$

D

$$2x^{2}- 14xy + 24y^{2}$$

E

$$-7x^{2}- 4xy + 20y^{2}$$

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