Chapter 5, Section 7

Chapter 5, Section 7

Factoring any old trinomial:

Look at the trinomial above. Each term is separated from the next by a plus sign.

caution:

In this course, we will be working with trinomials that have only integer coefficients so everything I teach you will apply only to this type of trinomial.

second caution:

Throughout much of this section I am assuming that the GCF is 1. By the end of the section you will have trinomials in which there is a real live GCF. You are always required to factor it out first before using any of these techniques!

If our trinomial is , then

a =

b =

c =

If our trinomial is

a =

b =

c =

What about ?

a =

b =

c =

Test for factorability. Once you’ve identified a, b, and c you can test to see if your trinomial factors nicely.

calculate

If the answer is a perfect square then the trinomial can be factored nicely.

If the answer is zero the trinomial has repeated factors.

If the answer is not a perfect square, you need techniques from College Algebra to factor it and the appropriate answer in this class is “prime”.

Let’s look at some trinomials and perform the test for factorability on them.

a = b = c =

=

Now we know that this trinomial can be factored by a student in this class.

Let’s look at

. Note that I can write this: . I can put in a home made term.

a = b = c =

=

Now we know that this trinomial can be factored by a student in this class.

Lets look at

a = b = c =

=

Now we know that this trinomial cannot be factored by a student in this class.

Let’s look at

a = b = c =

=

Now we know that this trinomial can be factored by a student in this class. And that is has a repeated factor.

Moving on to techniques of factoring.

Technique Afor use when a = 1

When a = 1, then both factors start with an x. For example: as opposed to .

Can be factored? How do you know?

= ( x + P) ( x + Q)

Can be factored? How do you know?

= ( 5x + P ) ( x + Q). We’ll use Technique B on this one!

So if a = 1 and the test for factorability works, we’ll do the following step:

Find two number that multiply to c and add to b. Put one number in as P and the other as Q.

Let’s work with = ( x + P) ( x + Q)

If I FOIL with the LHS I get … if I combine like terms I get

.

a = 1

b = P + Q

c = PQ

Hence my earlier instruction:

If a = 1, find two numbers, P and Q that multiply to c and add to b. Which is to say

P + Q = b and PQ = c.

Working with

Find two numbers, P and Q so that P + Q =  2 and PQ = 1.

So P = and Q =

and = ( x + ) ( x + )

Done…and exactly what the test for factorability said, too.

check your work

Let’s try with . We already did the test for factorability.

Find two numbers, P and Q so that P + Q = 3 and PQ = 2.

Use them in the following factors: ( x + ) ( x + ).

Summarizing Technique A

Identify the coefficients a, b, and c.

Calculate to see if the trinomial can be factored.

If a = 1, you know you have factors that look like ( x + P) ( x + Q).

Find two number, P and Q so that PQ = c and P + Q = b. Use them in the factors.

Let’s do it once more on .

a = 1b = c =

Find two numbers, P and Q, so that P + Q = b and PQ = c.

Use them in the factors ( x + P ) ( x + Q)

check your work
Factor .See it as

a = 1b = c =

Can this be factored by a student in this class?

If yes, continue. If no, write “prime” and go to the next problem.

In this case, note also that you know the sum of two squares cannot be factored.

Is there a repeated factor? Then you can write it as .

Find two numbers, P and Q, so that P + Q = b and PQ = c.

P + Q = ______PQ = ______

Put P and Q in the factors ( x + P ) ( x + Q )

check your work

Worksheet for Technique A

Factoring .

a = 1b = c =

Can this be factored by a student in this class?

If yes, continue. If no, write “prime” and go to the next problem.

Is there a repeated factor? Then you can write it as .

Find two numbers, P and Q, so that P + Q = b and PQ = c.

P + Q = ______PQ = ______

Put P and Q in the factors ( x + P ) ( x + Q )

check your work

Worksheet for Technique A

Factoring .

a = 1b = c =

Can this be factored by a student in this class?

If yes, continue. In no, write “prime”

Is there a repeated factor? Then you can write it as .

Find two numbers, P and Q, so that P + Q = b and PQ = c.

P + Q = ______PQ = ______

Put P and Q in the factors ( x + P ) ( x + Q )

check your work

Technique Bwhen a  1

Well, things are more complicated in this case. I’m going to depart from the book’s way and give you an alternate way. It is longer initially but once you get good at it, it always works.

FirstFactor out the GCF.

SecondCheck the non-GCF factor for factorability. If it will work, continue

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

Put P and Q in the temporary factors so you have

( ax + P ) ( ax + Q).

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer. Always.

Let’s do this:

We’ve seen that passes the factorability test.

3rdSet up an initial pair of factors( 5x + P) ( 5x + Q)

( note that these won’t really work because they actually multiply to )

4thfind two numbers, P and Q, so that

P + Q = b = 6 and PQ = ac = 5

5thset up temporary factors:( 5x + 1) (5x + 5)

factor out the 5 in the second factor and throw it away. What’s left is the answer:

( 5x + 1) ( x + 1)

6thcheck your work

Let’s do this again:

FirstFactor out the GCFHere, for this problem, the GCF is one.

SecondCheck the non-GCF factor for factorability. If it will work, continue.

a = b = c =

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

(6x + P) ( 6x + Q)

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

P + Q = b = 11 PQ = ac = 18

Put P and Q in the temporary factors so you have

( 6x + P ) ( 6x + Q).

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer.

SixthCheck by multiplying the binomials.

Lets look at

FirstFactor out the GCF.

SecondCheck the non-GCF factor for factorability.

a = b = c =

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

P + Q = PQ =

Put P and Q in the temporary factors so you have

( ax + P ) ( ax + Q) = ( 10x + _____) ( 10x + _____)

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer.

Sixthcheck your work

Let’s look at

FirstFactor out the GCF. Hint: This time there is one!

SecondCheck the non-GCF factor for factorability.

a = b = c =

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

P + Q = PQ =

Put P and Q in the temporary factors so you have

( ax + P ) ( ax + Q) = ( ____x + _____) ( ____x + _____)

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer.

Sixthcheck your work

Worksheet for Technique B when a  1.

FirstFactor out the GCF.

SecondCheck the non-GCF factor for factorability.

a = b = c =

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

P + Q = PQ =

Put P and Q in the temporary factors so you have

( ax + P ) ( ax + Q) = ( ____x + _____) ( ____x + _____)

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer.

Sixthcheck your work

Worksheet for Technique B when a  1.

FirstFactor out the GCF.

SecondCheck the non-GCF factor for factorability.

a = b = c =

ThirdSet up an initial pair of factors that will be modified as we go along.

( ax + P) ( ax + Q).

This will give you a TEMPORARY first term of .

FourthFind two numbers, P and Q, so that P + Q = b and PQ = ac.

P + Q = PQ =

Put P and Q in the temporary factors so you have

( ax + P ) ( ax + Q) = ( ____x + _____) ( ____x + _____)

Now here’s the important part,

FifthFactor out any common factor to each of these and THROW IT AWAY. What’s left is the answer.

SixthCheck your work

Practice – next lecture