Quandaries
and Queries 

Who is asking: Student Question: I also have a
question about factoring out common factors. In a problem like These are problems off my homework, but I want to know how to do them as well as just the answers. I'm a little behind my class in regards to quadratics and such, so I'd like to have a good understanding of it. Thanks so much! 

Hi, Factoring 12x^{2}+5x3 is like solving a mystery. There are lots of clues, you just have to see what they are. The major clues are the coefficient of x^{2} and the constant term, that is the 12 and the 3. The goal is to write
Lets look at the constant term first, the 3. From the right side above you can see that
Thus b and d are 1 and 3 but they have opposite signs since the product is 3. Hence this clue gives us
or
Now the second clue. Where does the 12 come from? In either of the two cases above
But 12 factors as
Now you need to do the "leg work". In each of the two possible factorizations of 12x^{2}+5x3 above you need to try
To see which one gives the coefficient of x as 5. For your second problem, x^{2 } 18x + 81, you are correct that if you attempt to factor out a 9 you get the coefficient of x^{2} as ^{1}/_{9}. But this isn't very nice as you want to work with integers, not fractions. Ther is no common factor here. Conside the clues as in the first problem. Since the coefficient of x^{2} is 1, I know that
Since the constant term is 81 I know that
Since 81 is positive, a and b have the same sign. But the coefficient of x is negative so
What are a and b so that a b = 81 and the coefficient of x is 18? Penny 

