Lesson Objectives

- Demonstrate an understanding of how to factor
- Demonstrate an understanding of how to multiply two binomials using FOIL
- Learn how to factor a trinomial with a leading coefficient of 1 into the product of two binomials

## How to Factor Trinomials when the Leading Coefficient is 1

Over the course of the last few lessons, we have learned some basic factoring techniques. So far, we have learned how to factor out the GCF from a polynomial and how to factor a four-term polynomial using the grouping method. In this lesson, we will learn how to factor a trinomial into the product of two binomials.

In most cases, our trinomials are of the form:

ax

This form is known as standard or general form. In this form, a represents the coefficient for x

The easiest scenario occurs when a, the coefficient of x

x

As an example, suppose we saw the following trinomial:

x

In this case, a (coefficient of x

(x + __)(x + __)

To fill in the blanks, we want to find two integers whose sum is b (10) and whose product is c (16). To do this we would list the factors of 16:

1,16

2,8

4,4

Of course, we could also have negatives involved, but these are not needed here since all signs are positive. We can see that the integers 2 and 8 fit our profile. They sum to 10 and have a product of 16. We can use those two integers to fill in the blanks. Note the order does not matter:

(x + 8)(x + 2) or (x + 2)(x + 8)

If we use FOIL to check this, we will get our trinomial (x

(x + 8)(x + 2):

F » x • x = x

O » 2 • x = 2x

I » 8 • x = 8x

L » 8 • 2 = 16

If we combine like terms we obtain our original trinomial:

x

Notice how the two coefficients of the O and I step (2 and 8) multiply to give us 16 and sum to 10.### Factoring 1x

Example 1: Factor each.

x

Step 1) Factor out any common factor other than 1 or -1

In this case, we don't have a common factor to pull out

Step 2) Set up two binomials where the first term in each is x

(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is 13, and c is 42. Let's list some factors of 42:

1, 42

2, 21

3, 14

6, 7

We can see that 6 and 7 are the desired integers.

6 + 7 = 13

6 • 7 = 42

Step 4) Use the two integers as the second terms for the binomials

(x + 6)(x + 7)

x

Example 2: Factor each.

4x

Step 1) Factor out any common factor other than 1 or -1

In this case, we can factor out a 4:

4(x

Step 2) Setup two binomials where the first term in each is x

4(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is -3, and c is -28. We know that a negative product comes from (+) • (-). We will list factors of 28 and then think about the signs:

1, 28

2, 14

4, 7

If we play with the signs, we can see that (-7) and (+4) are the desired integers:

-7 + 4 = -3

-7 • 4 = -28

Step 4) Use the two integers as the second terms for the binomials

4(x - 7)(x + 4)

4x

Example 3: Factor each.

2x

Step 1) Factor out any common factor other than 1 or -1

In this case, we can factor out a 2:

2(x

Step 2) Setup two binomials where the first term in each is x

2(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is -3, and c is -54. We know that a negative product comes from (+) • (-). We will list factors of 54 and then think about the signs:

1, 54

2, 27

3, 18

6, 9

If we play with the signs, we can see that (-9) and (+6) are the desired integers:

-9 + 6 = -3

-9 • 6 = -54

Step 4) Use the two integers as the second terms for the binomials

2(x - 9)(x + 6)

2x

In most cases, our trinomials are of the form:

ax

^{2}+ bx + cThis form is known as standard or general form. In this form, a represents the coefficient for x

^{2}, b represents the coefficient for x, and c represents our constant.The easiest scenario occurs when a, the coefficient of x

^{2}is 1. For this scenario, we could rewrite our trinomial as:x

^{2}+ bx + cAs an example, suppose we saw the following trinomial:

x

^{2}+ 10x + 16In this case, a (coefficient of x

^{2}) is 1, b (coefficient of x) is 10, and c (constant) is 16. Since a is 1, when we set up our binomials, we can start by placing an x in each first position:(x + __)(x + __)

To fill in the blanks, we want to find two integers whose sum is b (10) and whose product is c (16). To do this we would list the factors of 16:

1,16

2,8

4,4

Of course, we could also have negatives involved, but these are not needed here since all signs are positive. We can see that the integers 2 and 8 fit our profile. They sum to 10 and have a product of 16. We can use those two integers to fill in the blanks. Note the order does not matter:

(x + 8)(x + 2) or (x + 2)(x + 8)

If we use FOIL to check this, we will get our trinomial (x

^{2}+ 10x + 16) back:(x + 8)(x + 2):

F » x • x = x

^{2}O » 2 • x = 2x

I » 8 • x = 8x

L » 8 • 2 = 16

If we combine like terms we obtain our original trinomial:

x

^{2}+ 2x + 8x + 16 = x^{2}+ 10x + 16Notice how the two coefficients of the O and I step (2 and 8) multiply to give us 16 and sum to 10.

### Factoring 1x^{2} + bx + c:

- Factor out any common factor other than 1 or -1
- Set up two binomials where the first term in each is x
- Find two integers whose sum is b and whose product is c
- Use the two integers as the second terms for the binomials

Example 1: Factor each.

x

^{2}+ 13x + 42Step 1) Factor out any common factor other than 1 or -1

In this case, we don't have a common factor to pull out

Step 2) Set up two binomials where the first term in each is x

(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is 13, and c is 42. Let's list some factors of 42:

1, 42

2, 21

3, 14

6, 7

We can see that 6 and 7 are the desired integers.

6 + 7 = 13

6 • 7 = 42

Step 4) Use the two integers as the second terms for the binomials

(x + 6)(x + 7)

x

^{2}+ 13x + 42 = (x + 6)(x + 7)Example 2: Factor each.

4x

^{2}- 12x - 112Step 1) Factor out any common factor other than 1 or -1

In this case, we can factor out a 4:

4(x

^{2}- 3x - 28)Step 2) Setup two binomials where the first term in each is x

4(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is -3, and c is -28. We know that a negative product comes from (+) • (-). We will list factors of 28 and then think about the signs:

1, 28

2, 14

4, 7

If we play with the signs, we can see that (-7) and (+4) are the desired integers:

-7 + 4 = -3

-7 • 4 = -28

Step 4) Use the two integers as the second terms for the binomials

4(x - 7)(x + 4)

4x

^{2}- 12x - 112 = 4(x - 7)(x + 4)Example 3: Factor each.

2x

^{2}- 6x - 108Step 1) Factor out any common factor other than 1 or -1

In this case, we can factor out a 2:

2(x

^{2}- 3x - 54)Step 2) Setup two binomials where the first term in each is x

2(x + __)(x + __)

Step 3) Find two integers whose sum is b and whose product is c

In this case, b is -3, and c is -54. We know that a negative product comes from (+) • (-). We will list factors of 54 and then think about the signs:

1, 54

2, 27

3, 18

6, 9

If we play with the signs, we can see that (-9) and (+6) are the desired integers:

-9 + 6 = -3

-9 • 6 = -54

Step 4) Use the two integers as the second terms for the binomials

2(x - 9)(x + 6)

2x

^{2}- 6x - 108 = 2(x - 9)(x + 6)#### Skills Check:

Example #1

Factor each. $$x^{3}- 12x^{2}+ 27x$$

Please choose the best answer.

A

$$x(x - 3)(x - 9)$$

B

$$(x + 6)(x + 7)$$

C

$$(x + 3)(x - 9)$$

D

$$(x - 3)(x - 9)$$

E

$$3(x + 3)(x - 9)$$

Example #2

Factor each. $$4x^{3}+ 68x^{2}+ 288x$$

Please choose the best answer.

A

$$4(x + 8)(x + 7)$$

B

$$9(x + 7)(x + 3)$$

C

$$4x(x + 8)(x + 9)$$

D

$$4x(x + 18)(x + 4)$$

E

$$4x(x + 7)(x + 12)$$

Example #3

Factor each. $$5x^{2}- 15x - 350$$

Please choose the best answer.

A

$$5(x - 10)(x + 7)$$

B

$$5x(x + 10)(x - 7)$$

C

$$5(x + 2)(x - 35)$$

D

$$(x + 70)(x - 1)$$

E

$$(x + 35)(x - 10)$$

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