Learning Objectives
Introduction
Basically, when we factor, we reverse the process of multiplying the polynomial which was covered in Tutorial 6: Polynomials.
Tutorial
Step 2:
Divide
the GCF out of every term of the polynomial.
Example 1: Factor out the GCF: .
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.
Example 2: Factor out the 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.
Our GCF is (3x -1).
That is how we get the (x + 5) for our second ( ).
Factoring a Polynomial withStep 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.
Example 3: Factor by grouping: .
Example 4: Factor by grouping: .
(Where the number in front of x squared is 1)
Step 1: Set up a product
of two ( ) where each will hold two terms.
Step 2: Find the factors
that go in the first positions.
So it would look like this: (x )(x ).
Step 3: Find the
factors that go in the last positions.
As you are finding these factors, you have to
consider the sign of the
expressions:
If c is negative, your factors are going to have opposite signs depending on b’s sign.
Example 5: Factor the trinomial: .
So we go right into factoring the trinomial of the form .
(a )(a )
Putting that into our factors we get:
Example 6: Factor the trinomial: .
We need to factor out the GCF before we tackle the trinomial part of this.
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.
2y(x )(x )
Putting that into our factors we get:
(where a does not equal 1)
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.
Step 2: Use trial and
error to find the factors needed.
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.
Example 7: Factor the trinomial .
So we go right into factoring the trinomial of the form .
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 .
This is not our original polynomial.
So we need to try again.
Second try:
This is our original polynomial.
So this is the correct combination of factors for
this polynomial.
Example 8: Factor the trinomial .
So we go right into factoring the trinomial of the form
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 .
This is our original polynomial.
So this is the correct combination of factors for
this polynomial.
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.
Example 9: Factor the trinomial .
So we go right into factoring the trinomial of the form .
(x )(x )
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.
OR
This is the reverse of the binomial squared found in Tutorial 6: Polynomials. Recall that factoring is the reverse of multiplication.
Example 10: Factor the perfect square trinomial: .
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.
Example 11: Factor the perfect square trinomial:.
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.
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.
Example 12: Factor the difference of two squares: .
This fits the form of a the difference of two squares. So we will factor using that rule:
Example 13: Factor the difference of two squares: .
This fits the form of the difference of two squares. So we will factor using that rule:
Example 14: Factor the sum of cubes: .
This fits the form of the sum of cubes. So we will factor using that rule:
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.
Example 15: Factor the difference of cubes: .
This fits the form of the difference of cubes. So we will factor using that rule:
When you need to factor, you ALWAYS look for the GCF first. Whether 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 StrategyI. GCF:
II. Binomials:
b.
c.
III. Trinomials:
b. Trial and error:
c. Perfect square trinomial:
IV. Polynomials with four terms:
Example 16: Factor completely.
The GCF. In this case, there is one.
Factoring out the GCF of 3 we get:
Note that if we would multiply this out, we would get the original polynomial.
Example 17: Factor completely.
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:
Note that if we would multiply this out, we would get the original polynomial.
Example 18: Factor completely.
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:
Note that if we would multiply this out, we would get the original polynomial.
Example 19: Factor completely.
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:
Note that if we would multiply this out, we would get the original polynomial.
Example 20: Factor completely.
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:
Note that if we would multiply this out, we would get the original polynomial.
Practice Problems
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.
Practice Problems 1a - 1f: Factor completely.
Need Extra Help on these Topics?
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.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec. 13, 2009 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.