Question

Pierwiastkami wielomianu W(x) = x³ +px + q są liczby x1, x2, x3 spełniające warunki x2 = x1, x3 = x1 - 6
Oblicz log½(p+q)

+ wszystkie obliczenia;)

Answer 1

Załącznik.

Answer 2

W(x) = x³+px+q
W(x) = (x-x₁)(x-x₂)(x-x₃)

x₁=x₂
x₃=x₁-6

W(x) = (x-x₁)²(x-x₁+6) = (x²-2x₁x+x₁²)(x-x₁+6) =
= x³-x₁x²+6x²-2x₁x²+2x₁²x-12x₁x+x₁²x-x₁³+6x₁² =
= x³+(-x₁+6-2x₁)x²+(2x₁²-12x₁+x₁²)x-x₁³+6x₁² =
= x³+(6-3x₁)x²+(3x₁²-12x₁)x-x₁³+6x₁²

6-3x₁ = 0 <=> x₁ = 2
3x₁²-12x₁ = p <=> p = -12
-x₁³+6x₁² = q <=> q = 16

log½(p+q) = log½(4) = log½((½)⁻²) = -2