What is Factoring?

Date: 01/17/98 at 14:01:47
From: Melissa
Subject: Factoring

I do not get how to do this algebra. One of the sample questions is:  
x squared + 5x + 4

I don't understand how to start out factoring.

Date: 01/20/98 at 21:15:47
From: Doctor Loni
Subject: Re: Factoring

Factoring can be a very intimidating thing, but once you get the hang 
of it you will be a pro.

First, let me make sure you understand what factoring is. To factor 
anything, you get it into its smaller component parts, finding out 
what smaller parts make up the bigger part. For instance, when you 
factor a number, say 12, you break it into its smallest parts. You 
could start with 6 x 2. Then 6 can be factored into 2 x 3, so your 
factors of 12 could be 2 x 2 x 3. So to factor x^2 + 5x + 4 is to break 
it down into simpler parts.

You may have already done some problems like this:

   (x + 3)(x - 4)

Multiplying it out gives you:

    x^2 - 4x + 3x - 12

Simplifying:

    x^2 - x - 12

There are some very important things to notice here: you get the first 
term of the trinomial, x ^2 (they call it a trinomial because three
terms are being added) by multiplying the FIRST terms in each 
parenthesis (x * x). 
 
The last number in the trinomial, the constant (meaning it has no x's 
in it)  (-12) comes from multiplying the LAST  terms in each 
parenthesis (3  *  -4).

To get the middle term (the x term, which in this case is - x), first 
multiply the first term in the first parenthesis (x) by the last term in 
the second parenthesis (-4) (we can call these terms the OUTSIDE terms 
because they are the outside terms of the two parentheses). Then multiply 
the last term in the first parenthesis (3) by the first term in the 
second parenthesis (x) (we can call the terms in the second parenthesis 
the INSIDE terms because they are the inside terms of the two parentheses).
Then add the "inside" and the "outside" together. Thus the middle term 
becomes 3x - 4x = -x.

You can also remember this by remembering the word  F O I L.  Foil 
stands for First, Outside, Inside, Last.

To review:  in (x + 3)(x - 4)  the First is x* x or x^2,  the Outside 
is x * (-4)  (or -4x)  the Inside is3 * x  (or 3x)  and the Last is 
(3) * (-4) (or -12)

(x + 3) and (x - 4) are factors of x^2 - x - 12

Factoring a trinomial is just doing the reverse of what I just did in 
the problem above. You want to get from the trinomial back to the 
factors. First make sure your problem is written in the right way, 
that is, the term with x^2 comes first (the first term), the term with 
x (also called the middle term) comes next, and the constant - the term 
with no x in it - comes last. In your problem x^2 is the first term, 
5x is the middle term, and 4 is the last term.

Step 1 is to write the problem on your paper:

      x^2 + 5x + 4

Step 2 is to write two empty parentheses like this:

    (          )(         )     

We do this because we know the factors of a trinomial look this way.

Step 3 is to look at the first term, which is x^2. We know from the 
problem I did above that the x^2 term comes from multiplying the first 
term in the first parenthesis by the first term in the second 
parenthesis. What are the only two things that can be multiplied together 
to give you x^2?  That's right!  x * x will give you x^2. You write it 
like this:

       (x         )(x       )

Step 4 is to look at signs. Remember from the problem above that the 
constant term (the one with no x in it) is found by multiplying 
the last terms of each parenthesis. Because the sign of the constant 
term in your problem (4) is plus, the signs of both the numbers that 
multiply together to give you 4 have to be either positive or negative 
(because a + * + or a - * - gives you a positive number, and a + * - 
will give you a negative number). So your parentheses could now look 
like this:

       (x -  )(x -  )  or this:  (x +    )(x +  )

Look back at my previous problem to see what gave you a middle term.  
Remember you are adding the "outside" to the  "inside." If both signs 
are minus, this will give you a minus number; if both signs are plus 
it will give you a plus number. Because +5x is positive, you know that 
the signs have to be positive. Thus you get:

         (x +    )(x +    )

Step 5 is the tricky part. You know that the last term is found by 
multiplying the last two numbers in each parenthesis together. This 
means the numbers could either be 4 and 1, or 2 and 2 (because 
4 x 1 = 4 or 2 x 2 = 4). But you also know that the inside terms 
multiplied together plus the outside terms multiplied together will 
give you the middle term, so you have to try them out. Let's try 
2 and 2:

      (x+ 2)(x +2) = x^2 +2x +2x + 4  =  x^2 + 4x + 4

Whoops! The middle term is not right. We need 5x not 4x. Now let's 
try 4 and 1:

      (x + 4)(x + 1) = x^2 + 4x + 1x + 4  = x^2 + 5x +4

We got it. The factors of x^2 + 5x + 4 are (x + 4) and (x +1).

Let's try another one:  

   Factor  x^2 + x - 2

Here are our parentheses:

    (        ) (         )

Remember F O I L   (first, outside, inside, last)
x^2 is the first terms multiplied together and the only thing they can 
be are x and x:

     (x        )(x         )

The sign in front of the constant is a minus  (-2) so the signs have 
to be different (this is the only way we can get a -2, found by 
multiplying the last terms).
       
       (x +  ) (x -   )

Now, because the term in front of the x term (+x) is a plus we know 
that when we multiply the outside and inside terms together and add 
them to each other, we need to end up with a plus. Because the signs 
are different, the bigger term will have to be a plus. Now the only 
factors of 2 are 2 and 1 so they will be the only numbers we will  
have to worry about.

If we put 2 and 1 in like this:

     (x +2) (x -1) = x^2 -  x + 2x - 2 = x^2+ x - 2

If we had reversed the 2 and the 1 we would have ended up with -x 
instead of +x.

That was a long explanation. The main things to remember are FOIL 
(first, outside, inside, last) and that sometimes it may take a 
little trial and error to get the right facts. Always multiply the 
factors back out to make sure you end up with what you started with.

If you have more questions or need more help, let me know.
 
-Doctor Loni,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/