Aplicando los números combinatorios, desarrolla ?

Antes de empezar debemos conocer la combinatoria y su formula, porque la desarrollaremos para resolverla.

<img src="https://tex.z-dn.net/?f=%7Bm%20%5Cchoose%20n%7D%3D%20%5Cfrac%7Bm%21%7D%7Bn%21%28m-n%29%21%7D%20" />

Aplicando los números combinatorios, desarrolla :

a) (4x³ - 3y)⁴ =

Haremos uso de la combinatoria, iremos incrementando la potencia del primer elemento y reduciendo la del segundo.

<img src="https://tex.z-dn.net/?f=%284x%5E3-3y%29%5E4%3D%7B4%20%5Cchoose%200%7D%284x%5E3%29%5E4%2B%7B4%20%5Cchoose%201%7D%284x%5E3%29%5E3%28-3y%29%2B%7B4%20%5Cchoose%202%7D%284x%5E3%29%5E2%28-3y%29%5E2%20%20%5C%5C%20%5Chspace%2A%7B20mm%7D%20%20%20%2B%7B4%20%5Cchoose%203%7D%284x%5E3%29%28-3y%29%5E3%2B%7B4%20%5Cchoose%204%7D%28-3y%29%5E4%20%20%20%5C%5C%20%20%5C%5C%0A%3D%281%29%28256x%5E%7B12%7D%29%2B%284%29%2864x%5E%7B9%7D%29%28-3y%29%2B%286%29%2816x%5E6%29%289y%5E2%29%20%5C%5C%20%5Chspace%2A%7B20mm%7D%20%2B%284%29%284x%5E3%29%28-27y%5E3%29%2B%281%29%2881y%5E4%29%20%5C%5C%20%20%5C%5C%0A%3D256x%5E%7B12%7D-768x%5E9y%2B864x%5E6y%5E2-432x%5E3y%5E3%2B81y%5E4" />

b)<img src="https://tex.z-dn.net/?f=%20%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%2B4x%29%5E7%20" />

Resolvemos de modo similar

<img src="https://tex.z-dn.net/?f=%28%20%5Cfrac%7B-2%7D%7Bx%5E2%7D%20%2B4x%29%5E7%3D%7B7%20%5Cchoose%200%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E7%2B%7B7%20%5Cchoose%201%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E6%284x%29%2B%7B7%20%5Cchoose%202%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E5%284x%29%5E2%20%20%5C%5C%20%5C%5C%0A%5Chspace%2A%7B20mm%7D%20%20%20%7B7%20%5Cchoose%203%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E4%284x%29%5E3%2B%7B7%20%5Cchoose%204%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E3%284x%29%5E4%2B%7B7%20%5Cchoose%205%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%5E2%284x%29%5E5%20%5C%5C%20%20%5C%5C%0A%5Chspace%2A%7B20mm%7D%20%20%20%7B7%20%5Cchoose%206%7D%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%284x%29%5E6%2B%7B7%20%5Cchoose%207%7D%284x%29%5E7%0A%0A" />

<img src="https://tex.z-dn.net/?f=%5C%5C%20%5Chspace%2A%7B20mm%7D%20%20%20%3D%281%29%28%5Cfrac%7B-128%7D%7Bx%5E%7B14%7D%7D%29%2B%287%29%28%5Cfrac%7B64%7D%7Bx%5E%7B12%7D%7D%29%284x%29%2B%2821%29%28%5Cfrac%7B-32%7D%7Bx%5E%7B10%7D%7D%2916x%5E2%29%2B%2835%29%28%20%5Cfrac%7B16%7D%7Bx%5E8%7D%20%29%2864x%5E3%29%5C%5C%20%20%5C%5C%0A%20%5Chspace%2A%7B20mm%7D%20%20%20%2B%2835%29%28%5Cfrac%7B-8%7D%7Bx%5E6%7D%29%28256x%5E4%29%2B%2821%29%28%5Cfrac%7B4%7D%7Bx%5E4%7D%29%281024x%5E5%29%2B%287%29%28%5Cfrac%7B-2%7D%7Bx%5E2%7D%29%284096x%5E6%29%2B16384x%5E7%5C%5C%20%20%5C%5C%20%20%20%0A%20%5Chspace%2A%7B20mm%7D%20%20%20%3D-%5Cfrac%7B128%7D%7Bx%5E%7B14%7D%7D%2B%5Cfrac%7B-1792%7D%7Bx%5E%7B11%7D%7D-%5Cfrac%7B10752%7D%7Bx%5E8%7D%2B%5Cfrac%7B35840%7D%7Bx%5E5%7D-%5Cfrac%7B71680%7D%7Bx%5E2%7D%2B86016x%20%5C%5C%20%20%5C%5C%0A%5Chspace%2A%7B20mm%7D%20-57344x%5E4%2B16384x%5E7%0A%0A" />.