Question

The quotient when $1+x^2+x^4+x^6+\cdots +x^{34}$ is divided by $1+x+x^2+x^3+\cdots +x^{17}$

• A. $x^{17}-x^{15}+x^{13}-x^{11}+\cdots +x$
• B. $x^{17}+x^{15}+x^{13}+x^{11}+\cdots +x$
• C. $x^{17}+x^{16}+x^{15}+x^{14}+\cdots +1$
• D. $x^{17}-x^{16}+x^{15}-x^{14}+\cdots -1$

Answer 1

The correct option is D $x^{17}-x^{16}+x^{15}-x^{14}+\cdots -1$

Assuming
$(x^{17}+x^{16}+x^{15}+\cdots +1)\times (a{x}^{17}+b{x}^{16}+\cdots )=(1+x^2+x^4+x^6+.....+x^{32}+x^{34})$

Equating the coefficient of $x^{34}\text{and}{x}^{33},$ we get $a=1,$
$\text{and}b+a=0\Rightarrow b=-1$
Hence, only matching option is (d).