Question 982382
The polynomial you meant to write is {{{P(x)=x^9 + 2x^8 + 3x^7 +"..."+8x^2+9x}}} ,
where for each term the power of {{{x}}} and the coefficient add to {{{10}}} .
The remainder when a polynomial {{{P(x)}}} is divided by {{{(x-1)}}} is {{{P(1)}}} .
In this case {{{P(1)=1+2+3+"..."+8+9=highlight(45)}}}
I could add using pencil and paper, or a calculator,
but I know that sum to be the sum of {{{red(9)}}} consecutive terms of an arithmetic sequence,
with the first and last of those terms being {{{1}}} and {{{green(9)}}} .
So, {{{P(1)=1+2+3+"..."+8+green(9)=red(9)(1+green(9))/2=red(9)*10/2=9*5=highlight(45)}}} .