Solve Quartic with Casio FX-115MS

[Albert Chan]

(11-29-2018 05:28 PM)Albert Chan Wrote:  Example: f(x) = x^4 + 2 x^3 + 3 x^2 + 4 x + 5 = 0

Depress the Quartic (remove cubic term) by substitute x = y + k, where k = -b/(4a)
We get depressed quartic: y^4 + A y^2 + B y + C

Using Casio CALC, evaluate f(x) at y = 0, ±1 ...

Another way to calculate shifted polynomial coefficients is repeated synthetic divisions.
Shift x-axis to k = -2/(4*1) = -0.5

k> 1 2 3 4 5
k> 1 1.5 2.25 2.875 3.5625 <-- C
k> 1 1.0 1.75 2 <-- B
k> 1 0.5 1.5 <-- A

Repeated differentiations also work:

C = f(k) = k^4 + 2 k^3 + 3 k^2 + 4 k + 5 = 3.5625
B = f'(k) = 4k^3 + 6k^2 + 6k + 4 = 2
A = f''(k)/2! = (12k^2 + 12k + 6) / 2! = 1.5

As expected: y^3 coef. = (24k + 12) / 3! = 0, y^4 coef. = (24) / 4! = 1