- #1
julianwitkowski
- 133
- 0
Sorry if this is the wrong thread, seems appropriate but I'm kind of just learning math. This started as a thought experiment to learn some new technique. I've been trying to compute the total number of integer polynomials under certain restrictions: such that all coefficients, x, and the constant term are all among the variables within the count.
I was told that this can only be done by brute force/enumeration, but I feel like there must be a way otherwise.
Basically all polynomials must suit the following restrictions:
± ax³ ± bx² ± cx ± d = 0
Such That...
1 ≤ a ≤ 100 ∈ ℤ [a ≠ 0 since it's a 3rd degree]
0 ≤ b ≤ 100 ∈ ℤ
0 ≤ c ≤ 100 ∈ ℤ
0 ≤ d ≤ 100 ∈ ℤ
0 ≤ x ≤ 100 ∈ ℤ
1(1)³ + 1(1)² - 1(1) - 1 = 0 ... is an example that would be counted...
2(1)³ + 2(1)² - 2(1) - 1 = 0 ... is not an example because... 2(1)³ + 2(1)² - 2(1) - 1 = 1
___________________________________
Could I create a 6D space for ax³ + bx² + cx + d = n and focus on the root for n between -100 and 100 ?
___________________________________
Do you have any way I could approach this problem?
I was told that this can only be done by brute force/enumeration, but I feel like there must be a way otherwise.
Basically all polynomials must suit the following restrictions:
± ax³ ± bx² ± cx ± d = 0
Such That...
1 ≤ a ≤ 100 ∈ ℤ [a ≠ 0 since it's a 3rd degree]
0 ≤ b ≤ 100 ∈ ℤ
0 ≤ c ≤ 100 ∈ ℤ
0 ≤ d ≤ 100 ∈ ℤ
0 ≤ x ≤ 100 ∈ ℤ
1(1)³ + 1(1)² - 1(1) - 1 = 0 ... is an example that would be counted...
2(1)³ + 2(1)² - 2(1) - 1 = 0 ... is not an example because... 2(1)³ + 2(1)² - 2(1) - 1 = 1
___________________________________
Could I create a 6D space for ax³ + bx² + cx + d = n and focus on the root for n between -100 and 100 ?
___________________________________
Do you have any way I could approach this problem?