Determining what powers come out after polynomial multiplication Is there a quick method to determine what powers come out after polynomial multiplication? Specifically, I'm working with raising a polynomial by an integral power, so the binomial/multinomial theorem would be useful (though I have no idea how to use it).

For example, expanding out $(x+x^2+x^5)^3$ gives me:
$$=x^{15}+3x^{12}+3x^{11}+3x^9+6x^8+3x^7+x^6+3x^5+3x^4+x^3$$
Is there a quick way to know what powers come out (i.e. $15,12,11,\cdots$) just using the given powers of the unexpanded polynomial.

So for this case, the powers you can get are all the sums you can make using three terms (due to the power outside of the parenthesis) and the numbers 1,2, and 5 (the powers of the X's). 
So since we allow for a given number to  be repeated, there should be 10 different powers in the expansion (choose 3 terms from 3 choices with replacement):
$1+1+1=3$
$1+1+2=4$
$1+2+2=5$
$2+2+2=6$
$5+1+1=7$
$5+2+1=8$
$5+2+2=9$
$5+5+1=11$
$5+5+2=12$
$5+5+5=15$
