Question

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

• 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 $x^{17}-x^{16}+x^{15}-x^{14}+\cdots -1$

$1+x^2+x^4+\cdots +x^{34}=\frac{1-x^{36}}{1-x^2}$

$1+x+x^2+x^3+\cdots +x^{17}=\frac{1-x^{18}}{1-x}$

$⟹\frac{1+x^2+x^4+x^6+\cdots +x^{34}}{1+x+x^2+\cdots +x^{17}}=\frac{1-x^{36}}{1-x^{18}}\times \frac{1-x}{1-x^2}=\frac{1+x^{18}}{1+x}=x^{17}-x^{16}+x^{15}+\cdots -1$