Hacer este cociente notable[tex]x ^ 5 + 32 / x + 2[ / tex]?

Hola.

Te dejo los diferentes casos de cocientes notables para que puedas resolver tus deberes.

Caso 1 :

<img src="https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B%28x%5E%7Bn%7D-y%5E%7Bn%7D%29%7D%7B%28x-y%29%7D%7D%3Dx%5E%7Bn-1%7D%2Bx%5E%7Bn-2%7Dy%2Bx%5E%7Bn-3%7Dy%5E%7B2%7D%2B%5Cldots%20y%5E%7Bn-1%7D%7D" />

Caso 2 : (n debe ser par)

<img src="https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B%28x%5E%7Bn%7D-y%5E%7Bn%7D%29%7D%7B%28x%2By%29%7D%7D%3Dx%5E%7Bn-1%7D-x%5E%7Bn-2%7Dy%2Bx%5E%7Bn-3%7Dy%5E%7B2%7D-%5Cldots%20y%5E%7Bn-1%7D%7D" />

Caso 3 : (n debe ser impar)

<img src="https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B%28x%5E%7Bn%7D%2By%5E%7Bn%7D%29%7D%7B%28x%2By%29%7D%7D%3Dx%5E%7Bn-1%7D-x%5E%7Bn-2%7Dy%2Bx%5E%7Bn-3%7Dy%5E%7B2%7D-%5Cldots%20y%5E%7Bn-1%7D%7D" /> * Obs : <img src="https://tex.z-dn.net/?f=%20%7B%5Cfrac%20%7B%28x%5E%7Bn%7D%2By%5E%7Bn%7D%29%7D%7B%28x-y%29%7D%7D%20%3A%20%20No%20%5C%20%20es%20%5C%20un%5C%20%20cociente%5C%20notable" />

Ahora bien, para el ejercicio que has publicado, es de la forma del caso 3.

Cumpliendo la condición : n = 5 = impar, resolvemos :

<img src="https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%7B%5Cfrac%20%7B%28x%5E%7B5%7D%2B2%5E5%29%7D%7B%28x%2B2%29%7D%7D%3Dx%5E%7B5-1%7D-x%5E%7B5-2%7D%2A2%2Bx%5E%7B5-3%7D%2A2%5E%7B2%7D-x%5E%7B5-4%7D%2A2%5E%7B3%7D%2B2%5E4" />

<img src="https://tex.z-dn.net/?f=%5Cboxed%20%7B%20%7B%5Cfrac%20%7B%28x%5E%7B5%7D%2B2%5E5%29%7D%7B%28x%2B2%29%7D%7D%3Dx%5E%7B4%7D-2x%5E%7B3%7D%2B4x%5E%7B2%7D-8x%2B16%20%7D" />

Saludos!