Factoring Trinomials

Date: 04/13/2005 at 23:33:13
From: Jessica
Subject: about factoring Trinomials


I'm trying to factor d^2 - 33d - 280.  The answer should be in ( )( )
form.  I dont know how to get it to equal -33 but also have the two
numbers multiply to equal -280.

I've tried figuring out what two numbers add up to -33, and then tried 
to see if those numbers multiplied equaled 280 and all the numbers 
I've tried so far have not worked.

Date: 04/14/2005 at 10:00:54
From: Doctor Peterson
Subject: Re: about factoring Trinomials

Hi, Jessica.

I start by checking the signs: since -280 is negative, the two numbers 
we're looking for have opposite signs. Their sum, then, can be thought 
of as the _difference_ of two positive numbers whose product is 280. 
(We'll put a negative sign on the larger of those, so that the actual
sum is negative.)

Now, if your first few guesses don't yield an answer, we can try 
listing all the pairs of factors of 280, in an orderly fashion.  In
preparation for that, I will often find the prime factorization of the
number, which will help me decide which factors to try.  I get

  280 = 2*2*2*5*7

(I did that by thinking this way: try 2 first, and 280 = 2*140; that's 
still even, so 140 = 2*70; again, 70 = 2*35; and 35 = 5 * 7.  Each 
time I divided out a prime factor, I added it to my list.)

Now I know that no factor that is a multiple of 3 will work, so I can 
skip all those, as well as primes larger than 7. Here is a list of 
factor pairs:

  1 * 280
  2 * 140
  3 ... no, multiple of 3
  4 * 70
  5 * 56
  6 ... no, multiple of 3
  7 * 40
  8 * 35
  9 ... no, multiple of 3
 10 * 28
 11 ... no, the prime 11 isn't a factor
 12 ... no, multiple of 3
 13 ... no, the prime 13 isn't a factor
 14 * 20
 15 ... no, multiple of 3
 16 ... no, 8 is the biggest power of 2 that goes in
 17 ... no, the prime 17 isn't a factor
 18 ... no, multiple of 3
 19 ... no, the prime 19 isn't a factor

We can stop here, since when we get to 20 we will be reusing pairs we 
already have.

In this list you will find the pair you need, which differ by 33.  In 
reality, I would have stopped when I found that one; but if you go 
through the entire list like this and find nothing, then you know it 
CAN'T be factored!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/