Determining what powers come out after polynomial multiplication

Question

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).

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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.

Answer 1

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):

$1+1+1=3$

$1+1+2=4$

$1+2+2=5$

$2+2+2=6$

$5+1+1=7$

$5+2+1=8$

$5+2+2=9$

$5+5+1=11$

$5+5+2=12$

$5+5+5=15$

Answer 2

You may find the powers by picking one term from each parentheses $$(x+x^2+x^5)^3 =(x+x^2+x^5)(x+x^2+x^5)(x+x^2+x^5)$$

For example $(x^2)(x^5)(x^5)=x^{12}$ so you have some $x^{12}$ in your result.

The next question is how many $x^{12}$ we have?

You have to see in how many ways you can make $x^{12}$ by pocking one element from each parentheses.$$(2,5,5), (5,2,5),(5,5,2)$$ that is only $3$ ways, so we have $3x^{12}$ in our reslut.