. Dado p(x) = (x - 2) * (x + 2) * (x + 1) * (x ^ 2 - x + 1) y q(x) = x - 1 / 2 ; verificar en base a división sintética el valor del residuo r(x) y el cociente s(x)?

Primero modificaremos el vector p(x) para que la division sea mas sencilla

p(x) = (x - 2) * (x + 2) * (x + 1) * (<img src="https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20" /> - x + 1)

p(x) = <img src="https://tex.z-dn.net/?f=%20%28x%5E%7B2%7D%20%2B2x-2x-4%29%2A%28x%2B1%29%2A%28%20x%5E%7B2%7D%20-x%2B1%29%20%20" />

<img src="https://tex.z-dn.net/?f=p%28x%29%3D%20%28x%5E%7B2%7D-4%29%2A%28x%2B1%29%2A%28%20x%5E%7B2%7D%20-x%2B1%29%20" />

<img src="https://tex.z-dn.net/?f=p%28x%29%3D%20%28x%5E%7B3%7D%2B%20x%5E%7B2%7D%20-4x-4%29%20%2A%28%20x%5E%7B2%7D%20-x%2B1%29%20" />

<img src="https://tex.z-dn.net/?f=p%28x%29%3D%20%20x%5E%7B5%7D-%20x%5E%7B4%7D%2B%20x%5E%7B3%7D%20%2B%20x%5E%7B4%7D%20-%20%20x%5E%7B3%7D%2B%20x%5E%7B2%7D%20-4%20x%5E%7B3%7D%20%2B4%20x%5E%7B2%7D-4x-4x%5E%7B2%7D%20%2B4x-4%20" />

<img src="https://tex.z-dn.net/?f=p%28x%29%3D%20x%5E%7B5%7D%2B%20x%5E%7B2%7D%20-4%20x%5E%7B3%7D%20-4%20" />

Ahora dividimos los vectores p(x) / q(x).

La parte derecha de la división (el cociente)se coloca al final.

<img src="https://tex.z-dn.net/?f=x%5E%7B5%7D-4%20x%5E%7B3%7D%20%2B%20x%5E%7B2%7D-4%20" /> |x - 0.

5

<img src="https://tex.z-dn.net/?f=-%20x%5E%7B5%7D%20%2B%200.5%2Ax%5E%7B4%7D%20" />

_________________________

<img src="https://tex.z-dn.net/?f=0.5%2Ax%5E%7B4%7D-4%20x%5E%7B3%7D%2B%20x%5E%7B2%7D%20-4%20" />

<img src="https://tex.z-dn.net/?f=-0.5%2Ax%5E%7B4%7D%20%2B%200.25%2Ax%5E%7B3%7D" />

_________________________

[img = 10]

[img = 11]

_______________________

[img = 12]

[img = 13]

_______________________ - 0.

4375x - 4

0.

4375x - 0.

21875

_______________________ - 4.

21875

[img = 14]

Por lo tanto el residuo r(x) = - 4.

21875

El cociente s(x) = [img = 15].