SOLUTION: State the restrictions and then simplify: 12x^2-37x+21 ___________ _ <-fraction symbol 3x^2-13x+14 I need to simplify and find the restrictions. I do not know how to do

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Question 1126464: State the restrictions and then simplify:
12x^2-37x+21
_____________ <-fraction symbol
3x^2-13x+14
I need to simplify and find the restrictions. I do not know how to do this at all, and I need the answer asap
Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website!
12x^2-37x+21
x^2-37x+252. One can use guess and check, but this way of taking the constant in the first term and multiplying by the last term, factoring, then dividing the constants in the factor by the original number works well.
(x-28)(x-9) which becomes (x-(28/12)(x-(9/12)) or reduces to (x-(7/3)) and (x-(3/4))
the numerator factors are therefore (3x-7) and (4x-3)
3x^2-13x+14 factors into (3x-7)(x-2)
Therefore, the function will simplify into (4x-3)/(x-2)
The original function is not defined when the denominator is 0. That occurs when x=2
There is a hole in the function at the point x=(7/3), for even though the factors cancel out, there is a 0/0 result and x=exactly 7/3, and that is undefined. The graph won't show that.
x can be any other real number. There is a vertical asymptote at x=2

restrictions are where the denominator=0. The function is not defined there. Nor is it defined at the hole.