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Question 484399: Hello,the question I'm having trouble with is factoring polynomials completely. The equation is 2x^2+5x-12
word it would be two x squared plus five x minus twelve
Thank you for taking the time to read this email and for the help.
-Kaitlin
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Let's do the easy part first. The first term of your trinomial is  , so the first terms of your two binomial factors have to be  and  .
So far we have:
(x\ \mp\ \ \ ))
I'm using  and  because the only thing we know for sure about the two signs is that they are different. How do we know that? Because the constant term of the given trinomial is negative. We are going to have to find two numbers such that when they are multiplied together the result is -12. The only way to have a negative result when multiplying two numbers is for one of the numbers to be positive and the other negative.
The coefficient on the term is positive which means that multiplying times the number we don't know in the second binomial and adding that to the product of and the number we don't know in the first binomial has a positive result. The likelyhood, since is larger than , is that the plus sign is with the 2nd binomial -- so let's assume that for the moment and come back to this point if we don't get anywhere with that assumption.
Guess:
(x\ +\ \ \ ))
Now there are several ways to make 12: 1 times 12, 2 times 6, or 3 times 4, each pair considering the two possible ways to assign opposite signs. We need a result of +5.
Try 2 times 12 is 24 minus 1 times 1 is 1 : 23. Nope
Try 2 times 1 minus 1 times 12 is 12 : -11. Nope
Try 2 times 2 minus 1 times 6 : -2. Nope
Try 2 times 6 minus 1 times 2 : 10. Nope
Try 2 times 3 minus 1 times 4 : 2. Nope (but closer!)
Try 2 times 4 minus 1 times 3 : 5 Ah Ha!!!
(x\ +\ 4))
The next challenge you will face with factoring trinomials is trying to determine whether or not the trinomial is factorable over the rational numbers. You can go through the whole trial and error bit like above, go through every possibility without success, and then conclude that the trinomial is not factorable over the rationals.
Fortunately there is a much easier and less error prone way to go about making this determination.
Consider the quadratic trinomial:
If
is a perfect square, then the trinomial factors over the rational numbers.
Using your problem as an example:
(12)\ =\ 25\ +\ 96\ =\ 121\ =\ 11^2)
Which leads us to conclude that
is factorable over the rational numbers, and indeed we were able to find two such factors.
John
My calculator said it, I believe it, that settles it
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