I have a question about factorials....?

Question

If you have x^4/12! - x^5/15!

can you say that's = x^4/12!( 1 - x/3!)

If not, can someone factor this completely for me:

The goal is to factor is as efficiently as possible so there's as few multiplications as possible.

1 - x^3/3! + x^6/6! - x^9/9! + x^12/12! - x^15/15!

Answer 1

You can't do the "= x^4/12!( 1 - x/3!)" part because 15! isn't 12! * 3!.

I don't know how you'd go about factoring that though.

Answer 2

x^4/12! - x^5/15!

= x^4/12!(1-x/(13*14*15)) becuase 15! = 12! * 13* 14* 15

based on it

1 - x^3/3! + x^6/6! - x^9/9! + x^12/12! - x^15/15!

= 1- x^3/(1*2*3)(1- x^3/4*5*6(1- x^3/(7*8*9)(1- x^3/(10*11*12)(1- x^3/(13*14*15)))))  

Answer 3

Hint : 15! = 12! *13 *14 *15