Question

Let $r(x)$ be the remainder when the polynomial $x^{135}+x^{125}-x^{115}+x^5+1$ is divided by $x^3-x$. Then,

• A. $r(x)$ is the zero polynomial
• B. $r(x)$ is a non-zero constant
• C. degree of $r(x)$ is one
• D. degree of $r(x)$ is two

Answer 1

Answer: (C)

Let $p(x)=x^{135}+x^{125}-x^{115}+x^5+1$,
$q(x)=x^3-x$ and $p(x)=q(x)k+r(x)$
$x^{135}+x^{125}-x^{115}+x^5+1$
$=(x^3-x)k+a{x}^2+5x+c$
$[\because r(x)=a{x}^2+bx+c]$
Put $x=0$,
$\therefore c=1$
Put $x=1,3=a+b+c$
$\Rightarrow 3=a+b+1$
$\Rightarrow a+b=2...(i)$
Put $x=-1,-1=a-b+c$
$\Rightarrow -1=a-b+1$
$\Rightarrow a-b=-2...(ii)$
From Eqs. (i) and (ii), we get
$a=0,b=2$
$\therefore r(x)=2x+1$
$\therefore$ Degree of $r(x)=1$