What is arithmetic, geometric and harmonic means of the three roots of qubic polynomial x³ - ex² - x + π?

Question

What is
(a) arithmetic mean,
(b) geometric mean, and
(c) harmonic mean
of the three roots of qubic polynomial x³ - ex² - x + π?

Answer 1

if the roots are a, b, c then

P(x) = x³ - (a+b+c)x² + (ab+ac+bc)x - abc



So a+b+c = e

abc = -π

ab+ac+bc = -1

1/a + 1/b + 1/c = (ab+ac+bc)/abc = 1/π



arithmetic mean = (a+b+c)/3 = e/3

geometric mean = ³√abc = -³√π

harmonic mean = 3 / (1/a + 1/b + 1/c) = 3π

Answer 2

(y - a)(y - b)(y - c) = (y² - (a + b)y + ab)(y - c) = y³ - (a + b + c)y + (ab + ac + bc)y - abc



(a) arithmetic mean = e / 3

(b) geometric mean = -π^(1/3)

(c) harmonic mean = 3π

Answer 3

If you don't know i dont either