<s>[INST] Polynomials division algebra problem Find sum of coefficients of the quotient obtained in:
$$\frac{2x^n+x^{n-1}+x^{n-2}+...+x^2+x+5}{x-\frac{1}{2}}$$
I got "n" as the answer but according to the book is wrong, I don't know what is wrong exactly, but i want to know why the answer is "2n" ._. , thanks.
 [/INST] $$\frac{2x^n+x^{n-1}+x^{n-2}+...+x^2+x+5}{x-\frac{1}{2}}=$$
$$\frac{(x-\frac{1}{2})(2x^{n-1}+2x^{n-2}+\dots+2x^2+2x+2)+6}{x-\frac{1}{2}}=$$
$$2x^{n-1}+2x^{n-2}+\dots+2x^2+2x+2+\frac{6}{x-\frac{1}{2}}$$

So the quotient is: $2x^{n-1}+2x^{n-2}+\dots+2x^2+2x+2$
The remainder is: $6$
And the sum of coefficients in the quotient is: $\sum\limits_{i=0}^{n-1}2=2n$
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