problem z parametrem, dzielenie wielomianu

[monmon]

Oblicz dla jakich wartości parametrów \(\displaystyle{ a}\) i \(\displaystyle{ b}\) wielomian \(\displaystyle{ W(x) = x^{4} + ax^{3} - bx^{2} + 2x - 2}\) jest podzielny przez wielomian \(\displaystyle{ P(x)= x^{2} - x - 2}\)

[sea_of_tears]

\(\displaystyle{ x^2 -x-2 =(x+1)(x-2)
\newline
\begin{cases}
W(-1) = 0 \\
W(2)=0
\end{cases}}\)

[maweave]

\(\displaystyle{ P(x)= x^{2} - x - 2 = (x-2)(x+1) \\
W(2) = 0\\
W(-1) = 0\\
W(2) = 16 + 8a - 4b + 4 - 2 = 0\\
W(-1) = 1 - a - b - 2 -2 = 0\\
8a - 4b = - 18\\
a + b = -3 \\
a = -3 -b\\
8(-3-b) - 4b = -18\\
-24 - 8b - 4b = -18\\
- 12b = 6\\
b = -\frac{1}{2}\\
a = -\frac{5}{2}}\)