Question 662316: Factor completely, if the polynomial is prime, state this. 25a^a-23a-2. I do not understand these what so ever! I have tried using youtube to find videos and they always used little numbers or even. Can someone PLEASE HELP ME! You don't even have to tell me the answer just how to get the answer.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
You really mean  , then put a fork in yourself -- you are done. You aren't factoring this anytime in your lifetime.
On the other hand if you were just being sloppy and meant  , then read on.
You need to find two factors (you know it is only two factors because the exponent is 2) of the form  and  so that  ,  , and 
Quadratic trinomials come with three signs to consider, that is the sign on each of the three terms. In your case the sign on the first term is positive, and the sign on the other two is negative.
In order to get a negative sign on the constant term, the signs in the middle of each of your binomial factors must be OPPOSITE -- hence either  is a negative number or  is a negative number.
Now, let's get down to numbers. There are exactly three possibilities for the product  if we are dealing with only integer coefficients, namely 1 and 25, 5 and 5, or 25 and 1. There are only 2 possibilities to get the 2 as a constant term, 1 and 2 or 2 and 1.
Next, let's look at the linear or first degree term (the term in the middle of yor trinomial). We need -23, right. Well -25 plus 2 is exactly -23, isn't it?
So, how to make -25? Either -25 times 1 or 25 times -1. So let's try  and  .
Then if we let  and  , we end up with -25a plus 2a for the center term.
Use those numbers and FOIL it to check the work.
There are methods for systematically working these out, but I think they are a bit beyond where I think you are in your mathematics education. If you are interested, look up the Rational Roots Theorem and learn how to perform Synthetic Division -- you'll need both.
If you have a graphing calculator, then set your quadratic trinomial equal to zero and submit the equation to your calculator. If the calculator provides rational number zeros, then the quadratic is factorable, otherwise not. Furthermore, the factors are  and  where  and are the zeros of the function. If you end up with the zero being a rational number, i.e.  , then instead of writing your factor as  , you should write it  .
John
My calculator said it, I believe it, that settles it
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