Learning Objectives
Introduction
In this tutorial we add on to your factoring repertoire by talking about factoring trinomials. Basically, you will be doing the FOIL method backwards. This is one of those things that just takes practice to master. Make sure that you work through the problems on this page as well as any that you teacher may have assigned you. You never know when your math skills will be put to the test. Now let's get to factoring these trinomials.
Tutorial
(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.
So we go right into factoring the trinomial of the form .
(y )(y )
Putting that into our factors we get:
We need to factor out the GCF, as shown in Tutorial 27: The GCF and Factoring by Grouping, 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.
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 negatives here because the middle term is negative.
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.
So we go right into factoring the trinomial of the form
In the second terms of the binomials, we need factors of -6. This would have to be -6 and 1, 6 and -1, 3 and -2, or -3 and 2. 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 .
First try:
This is our original polynomial.
So this is the correct combination of factors for this polynomial.
Step 1: Substitute x for the
non-coefficient
part of the middle term.
When you substitute, the trinomial should be in the form OR .
If it isn't, then you may have to seek a different method than what we are covering here. All of the ones we are using substitution with on this web page will fit this form.
Step 2: If the trinomial
is in the form OR ,
factor it accordingly.
Step 3: Substitute back in what you replaced x with in step 1.
Since x is already being used in this problem, let's use y for our substitution.
*Substitute y in for x squared
Now we proceed as we did in examples 3 and 4 above.
In the second terms of the binomials, we need factors of -9. This would have to be -3 and 3, 9 and -1, or -9 and 1. 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 .
First try:
This is our original polynomial.
So this is the correct combination of factors for this polynomial.
*Substitute x in for (a + 2)
Now we proceed as we did in examples 1 and 2 above.
We need two numbers whose product is 32 and sum is -12. That would have to be -8 and -4.
Putting that into our factors we get:
*Combine like terms in ( )
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.
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 9 is 3 and 3, which does not
add up to be
2.
Another pair of factors are 1 and 9, which also does not
add up
to 2.
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.
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 - 1d: Factor Completely.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/factquad.htm
This webpage helps you factor trinomials.
http://www.mathpower.com/tut47.htm
This webpage helps you factor trinomials.
http://www.mathpower.com/tut31.htm
This webpage helps you factor trinomials.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 15, 2011 by Kim Seward.
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