Section7.3Factoring Trinomials with Leading Coefficient One
ΒΆIn Chapter 6, we learned how to multiply binomials like and obtain the trinomial In this section, we will learn how to undo that. So we'll be starting with a trinomial like and obtaining its factored form The trinomials that we'll factor in this section all have leading coefficient but Section 7.4 will cover some more general trinomials.
Subsection7.3.1Factoring Trinomials by Listing Factor Pairs
Consider the example There are at least three things that are important to notice:
The leading coefficient of is
The two factors on the right use the numbers and and when you multiply these you get the from
The two factors on the right use the numbers and and when you add these you get the from
So the idea is that if you need to factor and you somehow discover that and are special numbers (because and ), then you can conclude that is the factored form of the given polynomial.
Example7.3.2
Factor Since the leading coefficient is we are looking to write this polynomial as where the question marks are two possibly different, possibly negative, numbers. We need these two numbers to multiply to and add to How can you track these two numbers down? Since the numbers need to multiply to one method is to list all factor pairs of in a table just to see what your options are. We'll write every pair of factors that multiply to
We wanted to find all factor pairs. To avoid missing any, we started using as a factor, and then slowly increased that first factor. The table skips over using as a factor, because is not a factor of Similarly the table skips using and as a factor. And there would be no need to continue with and beyond, because we already found βlargeβ factors like as the partners of βsmallβ factors like
There is an entire second column where the signs are reversed, since these are also ways to multiply two numbers to get In the end, there are eight factor pairs.
We need a pair of numbers that also adds to All we need to do is check what each of our factor pairs add up to:
Factor Pair | Sum of the Pair |
(what we wanted) |
Factor Pair | Sum of the Pair |
(no need to go this far) | |
(no need to go this far) | |
(no need to go this far) | |
(no need to go this far) |
The winning pair of numbers is and Again, what matters is that and So we can conclude that
To ensure that we made no mistakes, here are two possible checks.
Multiply it Out
Multiplying out our answer should give us
We can also use a rectangular area diagram to verify the factorization is correct:
Evaluating
If the answer really is then notice how evaluating at would result in So the original expression should also result in if we evaluate at And similarly, if we evaluate it at should be
This also gives us evidence that the factoring was correct.
Example7.3.3
Factor The negative coefficient is a small complication from Example 7.3.2, but the process is actually still the same.
We need a pair of numbers that multiply to \(24\) and add to \(-11\text{.}\) Note that we do care to keep track that they sum to a negative total.
Factor Pair | Sum of the Pair |
\(1\cdot24\) | \(25\) |
\(2\cdot12\) | \(14\) |
\(3\cdot8\) | \(11\) (close; wrong sign) |
\(4\cdot6\) | \(10\) |
Factor Pair | Sum of the Pair |
\(-1\cdot(-24)\) | \(-25\) |
\(-2\cdot(-12)\) | \(-14\) |
\(-3\cdot(-8)\) | \(-11\) (what we wanted) |
\(-4\cdot(-6)\) | (no need to go this far) |
So \(y^2-11y+24=(y-3)(y-8)\text{.}\) To confirm that this is correct, we should check. Either by multiplying out the factored form:
\(y\) | \(-3\) | |
\(y\) | \(y^2\) | \(-3y\) |
\(-8\) | \(-8y\) | \(24\) |
Or by evaluating the original expression at \(3\) and \(8\text{:}\)
Our factorization passes the tests.
Example7.3.4
Factor The negative coefficient is again a small complication from Example 7.3.2, but the process is actually still the same.
We need a pair of numbers that multiply to \(-6\) and add to \(5\text{.}\) Note that we do care to keep track that they multiply to a negative product.
Factor Pair | Sum of the Pair |
\(1\cdot(-6)\) | \(-5\) (close; wrong sign) |
\(2\cdot(-3)\) | \(14\) |
Factor Pair | Sum of the Pair |
\(-1\cdot6\) | \(5\) (what we wanted) |
\(-2\cdot3\) | (no need to go this far) |
So \(z^2+5z-6=(z-1)(z+6)\text{.}\) To confirm that this is correct, we should check. Either by multiplying out the factored form:
\(z\) | \(-1\) | |
\(z\) | \(z^2\) | \(-z\) |
\(6\) | \(6z\) | \(-6\) |
Or by evaluating the original expression at \(1\) and \(-6\text{:}\)
Our factorization passes the tests.
Checkpoint7.3.5
Try one as an exercise.
Subsection7.3.2Connection to Grouping
The factoring method we just learned takes a bit of a shortcut. To prepare yourself for a more complicated factoring technique in Section 7.4, you may want to try taking the βscenic routeβ instead of that shortcut.
Example7.3.6
Let's factor again (the polynomial from Example 7.3.2). As before, it is important to discover that and are important numbers, because they multiply to and add to As before, listing out all of the factor pairs is one way to discover the and the
Instead of jumping to the factored answer, we can show how factors in a more step-by-step fashion using and Since they add up to we can write:
We have intentionally split up the trinomial into an unsimplified polynomial with four terms. In Section 7.2, we handled such four-term polynomials by grouping:
Now we can factor out each group's greatest common factor:
And we have found that factors as without memorizing the shortcut.
This approach takes more time, and ultimately you may not use it much. However, if you try a few examples this way, it may make you more comfortable with the more complicated technique in Section 7.4.
Subsection7.3.3Trinomials with Higher Powers
So far we have only factored examples of quadratic trinomials: trinomials whose highest power of the variable is However, this technique can also be used to factor trinomials where there is a larger highest power of the variable. It only requires that the highest power is even, that the next highest power is half of the highest power, and that the third term is a constant term.
In the four examples below, check:
if the highest power is even
if the next highest power is half of the highest power
if the last term is constant
Factor pairs will help withβ¦
Factor pairs won't help withβ¦
Example7.3.7
Factor This polynomial is one of the examples above where using factor pairs will help. We find that and so the numbers and can be used:
Actually, once we settled on using and we could have concluded that factors as if we know which power of to use. We'll always use half the highest power in these factorizations.
In any case, to confirm that this is correct, we should check by multiplying out the factored form:
Our factorization passes the tests.
Checkpoint7.3.8
Try one as an exercise.
Subsection7.3.4Factoring in Stages
Sometimes factoring a polynomial will take two or more βstagesβ. Always begin factoring a polynomial by factoring out its greatest common factor, and then apply a second stage where you use a technique from this section. The process of factoring a polynomial is not complete until each of the factors cannot be factored further.
Example7.3.9
Factor
We will first factor out the common factor, \(2\text{:}\)
Now we are left with a factored expression that might factor more. Looking inside the parentheses, we ask ourselves, βwhat two numbers multiply to be \(-40\) and add to be \(-3\text{?}\)β Since \(5\) and \(-8\) do the job the full factorization is:
Example7.3.10
Factor
The three terms don't exactly have a common factor, but as discussed in Section 7.1, when the leading term has a negative sign, it is often helpful to factor out that negative sign:
Looking inside the parentheses, we ask ourselves, βwhat two numbers multiply to be \(-24\) and add to be \(-2\text{?}\)β Since \(-6\) and \(4\) work here and the full factorization is shown:
Example7.3.11
Factor
First, always look for the greatest common factor: in this trinomial it is \(p^2q\text{.}\) After factoring this out, we have
Looking inside the parentheses, we ask ourselves, βwhat two numbers multiply to be \(-60\) and add to be \(4\text{?}\)β Since \(10\) and \(-6\) fit the bill, the full factorization can be shown below:
Subsection7.3.5More Trinomials with Two Variables
You might encounter a trinomial with two variables that can be factored using the methods we've discussed in this section. It can be tricky though: has two variables and it can factor using the methods from this section, but also has two variables and it cannot be factored. So in examples of this nature, it is even more important to check that factorizations you find actually work.
Example7.3.12
Factor This is a trinomial, and the coefficient of is so maybe we can factor it. We want to write where the question marks will be something that makes it all multiply out to
Since the last term in the polynomial has a factor of it is natural to wonder if there is a factor of in each of the two question marks. If there were, these two factors of would multiply to So it is natural to wonder if we are looking for where now the question marks are just numbers.
At this point we can think like we have throughout this section. Are there some numbers that multiply to and add to Yes, specifically and So we suspect that might be the factorization.
To confirm that this is correct, we should check by multiplying out the factored form:
Our factorization passes the tests.
In Section 7.4, there is a more definitive method for factoring polynomials of this form.
SubsectionExercises
1
Factor the given polynomial
2
Factor the given polynomial
3
Factor the given polynomial
4
Factor the given polynomial
5
Factor the given polynomial
6
Factor the given polynomial
7
Factor the given polynomial
8
Factor the given polynomial
9
Factor the given polynomial
10
Factor the given polynomial
11
Factor the given polynomial
12
Factor the given polynomial
13
Factor the given polynomial
14
Factor the given polynomial
15
Factor the given polynomial
16
Factor the given polynomial
17
Factor the given polynomial
18
Factor the given polynomial
19
Factor the given polynomial
20
Factor the given polynomial
21
Factor the given polynomial
22
Factor the given polynomial
23
Factor the given polynomial
24
Factor the given polynomial
25
Factor the given polynomial
26
Factor the given polynomial
27
Factor the given polynomial
28
Factor the given polynomial
29
Factor the given polynomial
30
Factor the given polynomial
31
Factor the given polynomial
32
Factor the given polynomial
33
Factor the given polynomial
34
Factor the given polynomial
35
Factor the given polynomial
36
Factor the given polynomial
37
Factor the given polynomial
38
Factor the given polynomial
39
Factor the given polynomial
40
Factor the given polynomial
41
Factor the given polynomial
42
Factor the given polynomial
43
Factor the given polynomial
44
Factor the given polynomial
45
Factor the given polynomial
46
Factor the given polynomial
47
Factor the given polynomial
48
Factor the given polynomial
49
Factor the given polynomial
50
Factor the given polynomial
51
Factor the given polynomial
52
Factor the given polynomial
53
Factor the given polynomial
54
Factor the given polynomial
55
Factor the given polynomial
56
Factor the given polynomial
57
Factor the given polynomial
58
Factor the given polynomial
59
Factor the given polynomial
60
Factor the given polynomial
61
Factor the given polynomial
62
Factor the given polynomial
63
Factor the given polynomial
64
Factor the given polynomial
65
Factor the given polynomial
66
Factor the given polynomial
67
Factor the given polynomial
68
Factor the given polynomial
69
Factor the given polynomial
70
Factor the given polynomial
71
Factor the given polynomial
72
Factor the given polynomial
73
Factor the given polynomial
74
Factor the given polynomial
75
Factor the given polynomial
76
Factor the given polynomial
77
Factor the given polynomial
78
Factor the given polynomial
79
Factor the given polynomial
80
Factor the given polynomial
81
Factor the given polynomial
82
Factor the given polynomial
83
Factor the given polynomial
84
Factor the given polynomial
85
Factor the given polynomial
86
Factor the given polynomial
87
Factor the given polynomial
88
Factor the given polynomial
89
Factor the given polynomial
90
Factor the given polynomial
91
Factor the given polynomial
92
Factor the given polynomial
93
Factor the given polynomial
94
Factor the given polynomial
95
Factor the given polynomial
96
Factor the given polynomial
97
Factor the given polynomial
98
Factor the given polynomial