SOLUTION: Factor the trinomial. m^2-mn-132n^2

Question 246709: Factor the trinomial.
m^2-mn-132n^2
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

The hardest part of factoring this is finding the factors of -132 that add up to -1. There may be a lot of factors of 132 and it might seem like it may take a lot of trial and error to figure out the right combination. But if we use some facts you know about multiplication and addition we can find the right combination relatively quickly:

-132 is a negative number and we are interested in finding the right pair of factors of -132 to use. To multiply a pair of numbers and get a negative result we must have one of the numbers positive and the other number negative.
The -1 coefficient in the middle is the result of adding the two factors together. We've already determined that one of the factors will be positive and one of these factors is negative. So we will add a positive number and a negative number and get -1 as a result.
So we want factors of -132 (one positive and one negative) that add up to -1. How do we add a positive and a negative number?:
- The result will have the sign of the "larger" number. (By "larger" I mean the number with the larger absolute value, of course.)
- Then subtract the positive version of the "larger" number minus the positive version of the "smaller" number. (These words sound complicated but I think you already know how to add a positive and a negative number.)
From this we can say that the negative factor must be "larger" because when we add the two factors of -132 the answer is negative. We can also say that the positive versions of both numbers are 1 apart from each other.
When two factors of a number are equal we call them square roots. We don't want equal factors in this problem. But we do want factors that are almost equal. We want them 1 apart from each other. Since and and , we want factors near 11 or 12.

So, with some logic we can avoid trying random pairs of factors of -132 until we happen to find the ones we need. Instead we can narrow our search to factors near 11 or 12 with one positive and one negative and the negative one being "larger". As it happens -132 = 11*(-12) so these are the factors we want. So:

You can check by multiplying out the right side and see if you get the left side as a result.
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