College Algebra Tutorial 7


College Algebra
Tutorial 7: Factoring Polynomials


WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Find the Greatest Common Factor (GCF) of a polynomial.
  2. Factor out the GCF of a polynomial.
  3. Factor a polynomial with four terms by grouping.
  4. Factor a trinomial of the form trinomial.
  5. Factor a trinomial of the form trinomial.
  6. Indicate if a polynomial is a prime polynomial.
  7. Factor a perfect square trinomial.
  8. Factor a difference of squares.
  9. Factor a sum or difference of cubes.
  10. Apply the factoring strategy to factor a polynomial completely.




desk Introduction


Factoring is to write an expression as a product of factors.  For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10.  We can also do this with polynomial expressions.  In this tutorial we are going to look at several ways to factor polynomial expressions.  By the time I'm through with you, you will be a factoring machine. 

Basically, when we factor, we reverse the process of multiplying the polynomial which was covered  in Tutorial 6: Polynomials.

 

 

desk Tutorial



 

Greatest Common Factor (GCF)
 
The GCF for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial.

  Factoring out the GCF
 
 
Step 1:  Identify the GCF of the polynomial.

 

Step 2:   Divide the GCF out of every term of the polynomial. 
 

This process is basically the reverse of the distributive property.

 
 
 

notebook Example 1:   Factor out the GCF: ex1a.

videoView a video of this example


 
Step 1:  Identify the GCF of the polynomial.

 
The largest monomial that we can factor out of each term is 2y.

 
Step 2:  Divide the GCF out of every term of the polynomial.

 
ex1b

*Divide 2y out of every term of the poly.

 
Be careful.  If a term of the polynomial is exactly the same as the GCF, when you divide it by the GCF you are left with 1, NOT 0.  Don’t think, 'oh I have nothing left', there is actually a 1.  As shown above when we divide 2y by 2y we get 1, so we need a 1 as the third term inside of the (  ).

Note that if we multiply our answer out, we should get the original polynomial.  In this case, it does check out.  Factoring gives you another way to write the expression so it will be equivalent to the original problem.


 
 
 

notebook Example 2:   Factor out the GCF: ex2a.

videoView a video of this example


 
This problem looks a little different, because now our GCF is a binomial. That is ok, we treat it in the same manner that we do when we have a monomial GCF.

Note that this is not in factored form because of the plus sign we have before the 5 in the problem.  To be in factored form, it must be written as a product of factors.


 
Step 1:  Identify the GCF of the polynomial.

 
This time it isn't a monomial but a binomial that we have in common. 

Our GCF is (3x -1).


 
Step 2:  Divide the GCF out of every term of the polynomial.

 
ex2b

*Divide (3x - 1) out of both parts

 
When we divide out the (3x - 1) out of the first term, we are left with x.  When we divide it out of the second term, we are left with 5. 

That is how we get the (x + 5) for our second (  ).


 

Factoring a Polynomial with 
Four Terms by Grouping
 
In some cases there is not a GCF for ALL the terms in a polynomial.  If you have four terms with no GCF, then try factoring by grouping.
 

Step 1: Group the first two terms together and then the last two terms together.

 

Step 2: Factor out a GCF from each separate binomial.

 

Step 3: Factor out the common binomial.


 
 
 

notebook Example 3:   Factor by grouping: ex3a.

videoView a video of this example


 
Note how there is not a GCF for ALL the terms.  So let’s go ahead and factor this by grouping.

 
Step 1: Group the first two terms together and then the last two terms together.

 
ex3b

*Two groups of two terms 

 
 
Step 2: Factor out a GCF from each separate binomial.

 
ex3c
*Factor out an x squared from the 1st (  )
*Factor out a 2 from the 2nd (  ) 

 
Step 3: Factor out the common binomial.

 
ex3d

*Divide (x + 3) out of both parts

 
Note that if we multiply our answer out, we do get the original polynomial.

 
 
 

notebook Example 4:   Factor by grouping: ex4a.

videoView a video of this example


 
Note how there is not a GCF for ALL the terms.  So let’s go ahead and factor this by grouping.

 
Step 1: Group the first two terms together and then the last two terms together.

 
ex4b

*Two groups of two terms 

 
Be careful.  When the first term of the second group of two has a minus sign in front of it, you want to put the minus in front of the second (   ).  When you do this you need to change the sign of BOTH terms of the second (  ) as shown above.

 
 
Step 2: Factor out a GCF from each separate binomial.

 
ex4c
*Factor out a  7x squared from the 1st (  )
*Nothing to factor out from the 2nd (  ) 

 
Step 3: Factor out the common binomial.

 
ex4d

*Divide (x - 2) out of both parts

 
Note that if we multiply our answer out that we do get the original polynomial.

 


 

Factoring Trinomials of the Form 

trinomial

(Where the number in front of x squared is 1)


 
Basically, we are reversing the FOIL method to get our factored form.  We are looking for two binomials that when you multiply them you get the given trinomial.

Step 1: Set up a product of two ( ) where each will hold two terms.
 

It will look like this:  (      )(      ).

 

Step 2: Find the factors that go in the first positions.
 
 

To get the x squared (which is the F in FOIL), we would have to have an x in the first positions in each (      ). 

So it would look like this:  (     )(x     ).

 

Step 3:  Find the factors that go in the last positions.
 
 

The factors that would go in the last position would have to be two expressions such that their product equals c (the constant) and at the same time their sum equals b (number in front of x term).

As you are finding these factors, you have to consider the sign of the expressions:
 

If c is positive, your factors are going to both have the same sign depending on b’s sign.

If c is negative, your factors are going to have opposite signs depending on b’s sign.


 

notebook Example 5:   Factor the trinomial: ex5a.

videoView a video of this example


 
Note that this trinomial does not have a GCF. 

So we go right into factoring the trinomial of the form trinomial.


 
Step 1: Set up a product of two ( ) where each will hold two terms.

 
It will look like this:       (      )(      )

 
Step 2: Find the factors that go in the first positions.

 
Since we have a squared as our first term, we will need the following:

(        )(      )


 
Step 3:  Find the factors that go in the last positions.

 
We need two numbers whose product is -14 and sum is -5.  That would have to be -7 and 2.

Putting that into our factors we get:


 
ex5b

*-7 and 2 are two numbers whose prod. is -14
and sum is -5

 
Note that if we would multiply this out, we would get the original trinomial.

 
 

notebook Example 6:   Factor the trinomial: ex6a.

videoView a video of this example


 
Note that this trinomial does have a GCF of 2y

We need to factor out the GCF  before we tackle the trinomial part of this.


 
ex6b

*Factor out the GCF of 2y

 
We are not finished, we can still factor the trinomial.  It is of the form trinomial.

Anytime you are factoring, you need to make sure that you factor everything that is factorable.  Sometimes you end up having to do several steps of factoring before you are done.


 
Step 1 (trinomial): Set up a product of two ( ) where each will hold two terms.

 
It will look like this:      2y(      )(      )

 
Step 2 (trinomial): Find the factors that go in the first positions.

 
Since we have x squared as our first term, we will need the following:

2y(        )(x       )


 
Step 3 (trinomial):  Find the factors that go in the last positions.

 
We need two numbers whose product is 15 and sum is 8.  That would have to be 5 and 3.

Putting that into our factors we get:


 
ex6c

*5 and 3 are two numbers whose prod. is 15
*and sum is 8

 
Note that if we would multiply this out, we would get the original trinomial.

 


 

Factoring Trinomials of the Form

trinomial

 (where a does not equal 1)


 
Again, this is the reverse of the FOIL method.

The difference between this trinomial and the one discussed above, is there is a number other than 1 in front of the x squared.  This means, that not only do you need to find factors of c, but also a.
 

Step 1:  Set up a product of two ( ) where each will hold two terms.
 

It will look like this (      )(      )

 

Step 2: Use trial and error to find the factors needed.
 
 

The factors of a will go in the first terms of the binomials and the factors of c will go in the last terms of the binomials. 

The trick is to get the right combination of these factors.  You can check this by applying the FOIL method.  If your product comes out to be the trinomial you started with, you have the right combination of factors.  If the product does not come out to be the given trinomial, then you need to try again.


 
 

notebook Example 7:   Factor the trinomial ex7a.

videoView a video of this example


 
Note that this trinomial does not have a GCF. 

So we go right into factoring the trinomial of the form trinomial.


 
Step 1:  Set up a product of two ( ) where each will hold two terms.

 
It will look like this:      (      )(      )

 
Step 2: Use trial and error to find the factors needed.

 
In the first terms of the binomials, we need factors of 3 x squared. This would have to be 3x and x.

In the second terms of the binomials, we need factors of 2.  This would have to be 2 and 1.  I used positives here because the middle term is positive.

Also, we need to make sure that we get the right combination of these factors so that when we multiply them out we get ex7a.


  Possible Factor Check using the FOIL Method First try:
ex7b








 

ex7c

This is not our original polynomial. 

So we need to try again.
   

 

 

Second try:
ex7d

 

 

 

 

 

ex7e




This is our original polynomial.

So this is the correct combination of factors for this polynomial.
 
 


 
This process takes some practice.  After a while you will get used to it and be able to come up with the right factor on the first try.

 
 
 

notebook Example 8:   Factor the trinomial ex8a.

videoView a video of this example


 
Note that this trinomial does not have a GCF. 

So we go right into factoring the trinomial of the form trinomial


 
Step 1:  Set up a product of two ( ) where each will hold two terms.

 
It will look like this:      (      )(      )

 
Step 2: Use trial and error to find the factors needed.

 
In the first terms of the binomials, we need factors of 5 x squared. This would have to be 5x and x.

In the second terms of the binomials, we need factors of -8.  This would have to be -8 and 1, 8 and -1, 2 and -4, or -2 and 4.  Since the product of these factors has to be a negative number, we need one positive factor and one negative factor.

Also we need to make sure that we get the right combination of these factors so that when we multiply them out we get ex8a.


  Possible Factors Check using the FOIL Method First try:
ex8b




ex8c

This is our original polynomial.

So this is the correct combination of factors for this polynomial.


 
 

Prime Polynomials
 
Not every polynomial is factorable.  Just like not every number has a factor other than 1 or itself.  A prime number is a number that has exactly two factors, 1 and itself.  2, 3, and 5 are examples of prime numbers. 

The same thing can occur with polynomials.  If a polynomial is not factorable we say that it is a prime polynomial.

Sometimes you will not know it is prime until you start looking for factors of it.  Once you have exhausted all possibilities, then you can call it prime.  Be careful.  Do not think because you could not factor it on the first try that it is prime.  You must go through ALL possibilities first before declaring it prime.


 

notebook Example 9:   Factor the trinomial ex9a.

videoView a video of this example


 
Note that this trinomial does not have a GCF. 

So we go right into factoring the trinomial of the form trinomial.


 
Step 1: Set up a product of two ( ) where each will hold two terms.

 
It will look like this:       (      )(      )

 
 
Step 2: Find the factors that go in the first positions.

 
Since we have x squared as our first term, we will need the following:

(        )(x       )


 
 
Step 3:  Find the factors that go in the last positions.

 
We need two numbers whose product is 12 and sum is 5.

Can you think of any???? 

Since the product is a positive number and the sum is a positive number, we only need to consider pairs of numbers where both signs are positive.

One pair of factors of 12 is 3 and 4, which does not add up to be 5.
Another pair of factors are 2 and 6, which also does not add up to 5.
Another pair of factors are 1 and 12, which also does not add up to 5.

Since we have looked at ALL the possible factors, and none of them worked, we can say that this polynomial is prime.  In other words, it does not factor.


 


 

Factoring a Perfect Square Trinomial

perfect square trinomial
OR
perfect square trinomial


 
It has to be exactly in this form to use this rule.  When you have a base being squared plus or minus twice the product of the two bases plus another base squared, it factors as the sum (or difference) of the bases being squared. 

This is the reverse of the binomial squared found in Tutorial 6: Polynomials.  Recall that factoring is the reverse of multiplication. 


 
 

notebook Example 10:   Factor the perfect square trinomial: ex10a.

videoView a video of this example


 
First note that there is no GCF to factor out of this polynomial. 

Since it is a trinomial, you can try factoring this by trial and error shown above.  But if you can recognize that it fits the form of a  perfect square trinomial, you can save yourself some time.


 
ex10b

*Fits the form of a perfect sq. trinomial
*Factor as the sum of bases squared

 
Note that if we would multiply this out, we would get the original polynomial.

 
 
 

notebook Example 11:   Factor the perfect square trinomial:ex11a.

videoView a video of this example


 
 
First note that there is no GCF to factor out of this polynomial. 

Since it is a trinomial, you can try factoring this by trial and error shown above.  But if you can recognize that it fits the form of a  perfect square trinomial, you can save yourself some time.


 
ex11b

*Fits the form of a perfect sq. trinomial
*Factor as the diff. of bases squared

 
Note that if we would multiply this out, we would get the original polynomial.

 
 


 

Factoring a Difference of Two Squares

difference


 
Note that the sum of two squares DOES NOT factor.

Just like the perfect square trinomial, the difference of two squares  has to be exactly in this form to use this rule.   When you have the difference of two bases being squared, it factors as the product of the sum and difference of the bases that are being squared.

This is the reverse of the product of the sum and difference of two terms found in Tutorial 6: Polynomials.  Recall that factoring is the reverse of multiplication. 


 
 
 

notebook Example 12:   Factor the difference of two squares: ex12a.

videoView a video of this example


 
First note that there is no GCF to factor out of this polynomial.

This fits the form of a the difference of two squares.  So we will factor using that rule:


 
ex12b

*Fits the form of a diff. of two squares
*Factor as the prod. of sum and diff. of bases

 
Note that if we would multiply this out, we would get the original polynomial.

 
 
 

notebook Example 13:   Factor the difference of two squares: ex13a.

videoView a video of this example


 
First note that there is no GCF to factor out of this polynomial.

This fits the form of the difference of two squares.  So we will factor using that rule:


 
ex13b

*Fits the form of a diff. of two squares
*Factor as the prod. of sum and diff. of bases

 
Note that if we would multiply this out and the original expression out we would get the same polynomial.


 

Factoring a Sum of Two Cubes

sum of cubes


 
The sum of two cubes  has to be exactly in this form to use this rule.  When you have the sum of two cubes, you have a product of a binomial and a trinomial.  The binomial is the sum of the bases that are being cubed.  The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared. 

 
 

notebook Example 14:   Factor the sum of cubes: ex14a.

videoView a video of this example


 
First note that there is no GCF to factor out of this polynomial.

This fits the form of  the sum of cubes.  So we will factor using that rule:


 
ex14b
*Fits the form of a sum of two cubes
*Binomial is sum of bases
*Trinomial is 1st base squared, minus prod. of bases, plus 2nd base squared

 
Note that if we would multiply this out, we would get the original polynomial.


 

Factoring a Difference of Two Cubes

difference of cubes


 
This is factored in a similar fashion to the sum of two cubes.  Note the only difference is that the sign in the binomial is a - which matches the original sign, and the sign in front of ax is positive, which is the opposite sign.

The difference of two cubes has to be exactly in this form to use this rule.  When you have the difference of two cubes, you have a product of a binomial and a trinomial.  The binomial is the difference of the bases that are being cubed.  The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared. 


 
 
 

notebook Example 15:   Factor the difference of cubes: ex15a.

videoView a video of this example


 
 
First note that there is no GCF to factor out of this polynomial.

This fits the form of  the difference of cubes.  So we will factor using that rule:


 
ex15b
*Fits the form of a diff. of two cubes
*Binomial is diff. of bases
*Trinomial is 1st base squared, plus prod. of bases, plus 2nd base squared

 
Note that if we would multiply this out, we would get the original polynomial.

 
 
Now that you have a list of different factoring rules, let’s put it all together.  The following is a checklist of the factoring rules that we have covered in our tutorials. 

When you need to factor, you ALWAYS look for the GCF firstWhether you have a GCF or not, then you continue looking to see if you have anything else that factors. 

Below is a checklist to make sure you do not miss anything.  Always factor until you can not factor any further.


 

Factoring Strategy

I.  GCF:
 

Always check for the GCF first, no matter what.

 

II.  Binomials:
 

a. difference

b. sum of cubes

c. difference of cubes

 

III. Trinomials:
 

a. trinomial

 

b. Trial and error:

trinomial

 

c.  Perfect square trinomial:

perfect square trinomial
perfect square trinomial


 

IV.  Polynomials with four terms:
 

Factor by grouping

 
 
 

notebook Example 16:   Factor ex16a completely.

videoView a video of this example


 
The first thing that we always check when we are factoring is WHAT?

The GCF.  In this case, there is one. 

Factoring out the GCF of 3 we get:


 
ex16b

*Factor a 3 out of every term

 
Next, we assess to see if there is anything else that we can factor.  We have a trinomial inside the (   ).  It fits the form of a perfect square trinomial, so we will factor it accordingly:

 
ex16c

*Fits the form of a perfect sq. trinomial
*Factor as the sum of bases squared

 
There is no more factoring that we can do in this problem.

Note that if we would multiply this out, we would get the original polynomial.


 
 
 

notebook Example 17:   Factor ex17a completely.

videoView a video of this example


 
The first thing that we always check when we are factoring is WHAT?

The GCF.  In this case, there is not one. 

So we assess what we have. It fits the form of a difference of two squares, so we will factor it accordingly:


 
ex17b

*Fits the form of a diff. of two squares
*Factor as the prod. of sum and diff. of bases

 
Next we assess to see if there is anything else that we can factor.  Note how the second binomial is another difference of two squares.  That means we have to continue factoring this problem.

 
ex17c

*Fits the form of a diff. of two squares
*Factor as the prod. of sum and diff. of bases

 
There is no more factoring that we can do in this problem.

Note that if we would multiply this out, we would get the original polynomial.


 
 
 

notebook Example 18:   Factor ex18a completely.

videoView a video of this example


 
The first thing that we always check when we are factoring is WHAT?

The GCF.  In this case, there is not one. 

So we assess what we have. It fits the form of a sum of two cubes, so we will factor it accordingly:


 
ex18b

*Fits the form of a sum of two cubes
*Binomial is sum of bases
*Trinomial is 1st base squared, minus prod. of bases, plus 2nd base squared

 
There is no more factoring that we can do in this problem.

Note that if we would multiply this out, we would get the original polynomial.


 
 
 

notebook Example 19:   Factor ex19a completely.

videoView a video of this example


 
The first thing that we always check when we are factoring is WHAT?

The GCF.  In this case, there is not one. 

So we assess what we have. This is a trinomial that does not fit the form of a perfect square trinomial.  Looks like we will have to use trial and error:


 
ex19b

*Factor by trial and error

 
There is no more factoring that we can do in this problem.

Note that if we would multiply this out, we would get the original polynomial.


 
 
 

notebook Example 20:   Factor ex20a completely.

videoView a video of this example


 
The first thing that we always check when we are factoring is WHAT?

The GCF.  In this case, there is not one. 

So we assess what we have. This is a polynomial with four terms.  Looks like we will have to try factoring it by grouping:


 
ex20b

*Group in two's
*Factor out the GCF out of each separate (   )
*Factor out the GCF of (x + 5b)

 
There is no more factoring that we can do in this problem.

Note that if we would multiply this out, we would get the original polynomial.

 

 

desk Practice Problems


These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1f: Factor completely.


 
1a. prob1a
(answer/discussion to 1a)
1b. prob1b
(answer/discussion to 1b)

 
1c. prob1c
(answer/discussion to 1c)
1d. prob1d
(answer/discussion to 1d)

 
1e. prob1e
(answer/discussion to 1e)
1f. prob1f
(answer/discussion to 1f)

 

 

desk Need Extra Help on these Topics?


The following are webpages that can assist you in the topics that were covered on this page:
 
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut27_gcf.htm
This webpage goes over how to factor out a GCF and how to factor by grouping.

http://www.purplemath.com/modules/simpfact.htm
This webpage helps you with factoring out the GCF.

http://www.mathpower.com/tut111.htm 
This webpage will help you with factoring out the GCF. 

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.htm
This webpage goes over how to factor trinomials.

http://www.purplemath.com/modules/factquad.htm
This webpage helps you factor trinomials.

http://www.mathpower.com/tut47.htm
This website helps you factor trinomials.

http://www.mathpower.com/tut31.htm
This website helps you factor trinomials.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/ int_alg_tut29_specfact.htm
This webpage goes over how to factor perfect square trinomial, difference of squares, and sum or difference of cubes.

http://www.sosmath.com/algebra/factor/fac05/fac05.html
This webpage helps you with the factoring by special products discussed in this tutorial.

http://www.purplemath.com/modules/specfact.htm
This webpage helps you with the factoring by special products discussed in this tutorial.


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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