Final answer: approx -902.9694830736585
For easier typing, let the roots be r,s,t
P(x) = ax^3 + bx^2 + cx + d
Find: (r+s)/t + (r+t)/s + (s+t)/r
Vieta's Formulas:
-b/a = 35;
c/a = rs + rt + st
-d/a = rst
b = -35a
P(x) = a*x^3 - 35*a*x^2 + c*x + d
P(k) - P(-k) = 2(a*k^3 + c*k)
P(0) = d
1/3: 2(a/27 + c/3) = 8d
a + 9c = 108d
1/4: 2(a/64 + c/4) = 9d
a + 16c = 288d
If we assume a=1:
P(x) = x^3 - 35*x^2 + c*x + d
108d - 9c = 1
288d - 16c = 1
c = -5/24
d = -7/864
P(x) = x^3 - 35*x^2 - (5/24)*x - 7/864
Get rid of the fractions; multiply all coefficients by 864
P(x) = 864x^3 - 30240*x^2 - 180*x - 7
Use Python console to check the given constraints, both successful:
In [17]: (p(1/3) - p(-1/3))/p(0)
Out[17]: 8.0
In [18]: (p(1/4) - p(-1/4))/p(0)
Out[18]: 9.0
Use Wolfram Alpha to find the roots of the cubic equation:
r ≈ (approx) 35.006
s ≈ -0.002979 + 0.014919 i
t ≈ -0.002979 - 0.014919 i
Sum of the roots checks = 35 (at least approx)
Use Python console, which does complex math:
r = 35.006
s = complex(-0.002979, 0.014919)
t = complex(-0.002979, -0.014919)
Find the desired quantity
In [19]: (r+s)/t + (r+t)/s + (s+t)/r
Out[19]: (-902.9694830736585+0j)
The desired value has no complex component:
approximately -902.9694830736585
|
|
Posted by Larry
on 2023-12-09 10:18:33 |
Analytic Solution with help from Wolfram
