Posted: Fri May 06, 2005 11:16 pm    Post subject: Factorization (Urgent)

Given any integer a and b. Can f(x) = (ax+b)x(x+1)(x-1) + 1 be factorized? (Only consider integral coefficients) I have no ideas to deal with the a and b… Can anyone please give me a hand? Please, I only want some hints. This is my assignment and I have to hand in the solution tomorrow.

Thank you.

Posted: Sat May 07, 2005 12:06 am    Post subject:

I can find 3 conditions that f(x) can be factorize. f(-1) = 1 f(0) = 1 f(1) = 1 Therefore, all the possible factor of f(x) with degree 2 : -x[sup]2[/sup] – x + 1 -x[sup]2[/sup] + x + 1 -2x[sup]2[/sup] + 1 Then expand f(x) to get f(x) = ax[sup]4[/sup] + bx[sup]3[/sup] – ax[sup]2[/sup] – bx + 1 Use f(x) to subtract each of the above possible factor. Then I can get three case. And they all should be divisible by the corresponding factor if the possible factors are really factors of f(x). I can get the three solutions that f(x) can be factorized are: 1. a = 1 b = 2 2. a = 3 b = -1 3. a = 4 b = 0 But I think I can still have more…. Maybe I can get some linear factors that are factors of f(x)…. Or it’s all the possible case already? (Since the factors will degree 2 already contained the linear factors?) The number inside[sup][/sup] is the degree.

I am sorry about that it cannot be shown in the right way.