Wielomian W(x)=x^3+ax^2+(b-1)x+2c jest podzielny przez trójmian Q(x)=x^2+x-2. Reszta z dzielenia wie.... Question from @Deme

January 2019 1 28 Report

Wielomian W(x)=x^3+ax^2+(b-1)x+2c jest podzielny przez trójmian Q(x)=x^2+x-2. Reszta z dzielenia wielomianu W(x) przez dwumian x+3 jest równa 4. Wyznacz współczynniki a,b,c oraz podaj pierwiastki wielomianu W(x).

Odpowiedzi : a=5,b=3,c=-4 pierwiastki to -4,-2,1

ghe $W(x)=x^3+ax^2+(b-1)x+2c$

$Q(x)=x^2+x-2$

$\Delta=1^2-4 \cdot 1 \cdot (-2)=1+8=9$

$\sqrt{\Delta}=\sqrt{9}=3$

$x_1= \frac{-1-3}{2}=\frac{-4}{2}=-2$

$x_2= \frac{-1+3}{2}=\frac{2}{2}=1$

Z warunków podanych w zadaniu mamy:
$\begin{cases}W(-2)=0\\ W(1)=0\\W(-3)=4\end{cases}$

$\begin{cases}(-2)^3+a \cdot (-2)^2+(b-1) \cdot (-2)+2c=0\\ 1^3+a \cdot 1^2+(b-1) \cdot 1+2c=0\\(-3)^3+a \cdot (-3)^2+(b-1) \cdot (-3)+2c=4\end{cases}$

$\begin{cases}-8+4a -2b+2+2c=0\\ 1+a+b-1+2c=0\\-27+9a -3b+3+2c=4\end{cases}$

$\begin{cases}4a-2b+2c=6\ /:2\\ a+b+2c=0\\9a-3b+2c =28\end{cases}$

$\begin{cases}2a-b+c=3\\ a+b+2c=0\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2a+c-3\\a+b+2c=0\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2a+c-3\\ a+2a+c-3+2c=0\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2a+c-3\\ 3a+3c=3\ /:3\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2a+c-3\\ a+c=1\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2a+c-3\\ a=1-c\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2(1-c)+c-3\\ a=1-c\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=2-2c+c-3\\ a=1-c\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\9a-3b+2c =28\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\9(1-c)-3(-c-1)+2c =28\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\9-9c+3c+3+2c =28\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\-9c+3c+2c =28-9-3\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\-4c =16\ /:(-4)\end{cases}$

$\begin{cases}b=-c-1\\ a=1-c\\c =-4\end{cases}$

$\begin{cases}b=4-1\\ a=1+4\\c =-4\end{cases}$

$\begin{cases}a=5\\ b=3\\c =-4\end{cases}$
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$W(x)=x^3+5x^2+2x-8$

$\begin{array}{lll} (x^3+5x^2+2x-8) & : & (x^2+x-2) =x+4\\ \underline{-x^3-x^2+2x} & \\ \qquad {4x^2 +4x -8 & & \\ \ \ \ \ \underline{-4x^2 - 4x+8} & &\\ \quad \qquad \qquad \qquad0 & & \end{array}$
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Pierwiastki wielomianu to : -4;-2; 1

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