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:
ax2 + bx + c
This form is known as standard or general form. In this form, a represents the coefficient for x2, b represents the coefficient for x, and c represents our constant.
The easiest scenario occurs when a, the coefficient of x2 is 1. For this scenario, we could rewrite our trinomial as:
x2 + bx + c
As an example, suppose we saw the following trinomial:
x2 + 10x + 16
In this case, a (coefficient of x2) 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 (x2 + 10x + 16) back:
(x + 8)(x + 2):
F » x • x = x2
O » 2 • x = 2x
I » 8 • x = 8x
L » 8 • 2 = 16
If we combine like terms we obtain our original trinomial:
x2 + 2x + 8x + 16 = x2 + 10x + 16
Notice how the two coefficients of the O and I step (2 and 8) multiply to give us 16 and sum to 10.

### Factoring 1x2 + 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
Let's take a look at a few examples.
Example 1: Factor each.
x2 + 13x + 42
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)
x2 + 13x + 42 = (x + 6)(x + 7)
Example 2: Factor each.
4x2 - 12x - 112
Step 1) Factor out any common factor other than 1 or -1
In this case, we can factor out a 4:
4(x2 - 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)
4x2 - 12x - 112 = 4(x - 7)(x + 4)
Example 3: Factor each.
2x2 - 6x - 108
Step 1) Factor out any common factor other than 1 or -1
In this case, we can factor out a 2:
2(x2 - 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)
2x2 - 6x - 108 = 2(x - 9)(x + 6)

#### Skills Check:

Example #1

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

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

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

A
$$5(x - 10)(x + 7)$$
B
$$5x(x + 10)(x - 7)$$
C
$$5(x + 2)(x - 35)$$
D
$$(x + 70)(x - 1)$$
E
$$(x + 35)(x - 10)$$