Roots of equation

Algebra Level 3

Let $z_1, z_2, z_3, z_4$ be the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$ . Find the least possible value of $\big \lfloor | z_1 + z_2 | \big \rfloor + 1$ .

Notations:

$| \cdot |$ denotes the absolute value function .
$\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 1.

1 solution

Fletcher Mattox
Jun 17, 2020

While this quartic is easy to solve:

$(z - 1)(z^4 + z^3 + z^2 + z + 1 = z^5 -1 = 0$

$z^5 = 1 \quad z \neq 1$

I prefer to write code:

from sympy import *
from sympy.abc import x, y
from itertools import combinations

roots = solve(x**4 + x**3 + x**2 + x + 1)
kmin = 100
for i,j in combinations(roots, 2):
    k = abs(i+j)
    if k < kmin:
        kmin = k

print("minimum:", kmin) 
print("solution:", (floor(kmin)+1))

# minimum: -1/2 + sqrt(5)/2
# solution: 1

Roots of equation

The answer is 1.

1 solution

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