Which of the expressions will have the larger coefficient for the x20 term after expanding and collecting terms? Might they possibly be the same?
( 1 + x2 - x3 )1000 or ( 1 - x2 + x3 )1000
The coefficient of x20 in ( 1 + x2 - x3 )1000 is greater than that in ( 1 - x2 + x3 )1000.
Experiment with the expansions by first squaring, cubing, etc. each polynomial.
It is possible to demonstrate that the absolute values of the coefficients of both expressions will be the same. However, note that the coefficients of the odd powers of x in the expansions change signs and the coefficients of the even powers do not. In fact, the positive powers in the expansions only yield positive terms to the sum.
We may assume that the coefficients of the second expression ( 1 - x2 + x3 )1000 will have some negative terms, but the first polynomial will remain all positive. Thus, ( 1 + x2 - x3 )1000 is certain to have the larger coefficint for x20.
