The Steps of Factoring

Date: 08/08/98 at 10:53:10
From: Brandie
Subject: Factoring problems

My question concerns the following problems:

Given y^2 - 14y + 24

   a. write factors of the first term: (______)(_______)

   b. signs are ________________ because the sign of last term of the
      trinominal is positive and the sign of the middle term is
      _________________________: (________________)(________________)

   c. find factors of 24 whose sum is -14: _____________and___________

   d. complete:(___________________)(______________________)

I am also have trouble factoring:

   a^2 + 25a + 24 = (a+1)(a+_______)

   y^2 - 12y + 27 = (a-3)(a-_______)

Please also can you tell me where to go and look for more help on 
factoring in Algebra I.

Date: 08/14/98 at 13:24:01
From: Doctor Margaret
Subject: Re: factoring problems

Hi Brandie,

Thanks for writing to Dr. Math. It looks as if you are just starting to 
learn to factor. This is a very nice way of giving a factoring problem 
because it lists all the things you have to do. It might help for you 
to remember that the factors of a number are two numbers that when 
multiplied together give you the number. For example, the factors of 
12 are:

   1 * 12
   2 * 6
   3 * 4

In factoring a trinomial we want to get the numbers that, when 
multiplied together using the FOIL method, give us back the trinomial.  
There is a method for this. First, we find out the factors of the 
first term in the trinomial.

The factors of y^2 (that's how we write y squared) are y * y.

There are two signs in the trinomial between the three terms. The sign 
on the last term gives us a clue about both of the signs within the 
parentheses. It would be good if you could memorize:

 - If the sign on the last term is positive, both signs in the 
   parentheses are the same.
 
 - If the sign on the last term is negative, then the signs in the 
   parentheses are different.

 - If the signs are the same, the first sign will tell you if they are 
   positive or negative. 

In your first example, we know that both parentheses will contain 
negative signs.  

Now let's look at the factors of 24 that will add up to -14. Remember 
that factors are numbers which in this case multiply together to get 24 
and add to get -14.  The factors of 24 are:

   1 * 24
   2 * 12
   3 * 8
   4 * 6

If we make 2 * 12 into -2 * -12 these numbers multiply to 24 and add 
to -14:

   (-2)*(-12) = 24
   (-2)+(-12) = -14

Our final answer is:

   y^2 -14y + 24 = (y -2)(y - 12)

We know that this is true because, using FOIL, we get:

   First: y * y = y^2
   Outer: -12 * y = -12y
   Inner: -2 * y = -2y
   Last: (-2)*(-12) = 24

So: y^2 -12y -2y + 24 = y^2 - 14y + 24, which is what we started with.

Try to do the next two problems by yourself. It would be a good idea 
to practice a lot. It does get easier, and you will be using it for a 
long time, so you have to get good at it. There are also some problems 
in our Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/faq.learn.factor.html   

Feel free to write back if you are still having trouble, and try to 
include any work you've done so we can see exactly where you are stuck.

- Doctor Margaret, The Math Forum
Check out our web site! http://mathforum.org/dr.math/