Q29 / 54 · KVPY 2017 (5 Nov SA) · Mathematics

Let $r(x)$ be the remainder when the polynomial $x^{135}+x^{125}-x^{185}+x^{5}+1$ is divided by $x^{3}-x .$ Then

  • A
    $r(x)$ is the zero polynomial
  • B
    $r(x)$ is a nonzero constant
  • Correct answer
    C
    degree of $r(x)$ is one
  • D
    degree of $r(x)$ is two

Solution

$\mathrm{x}^{135}+\mathrm{x}^{125}-\mathrm{x}^{115}+\mathrm{x}^{5}+1=\mathrm{k}\left(\mathrm{x}^{3}-\mathrm{x}\right)+\mathrm{A} \mathrm{x}^{2}+\mathrm{Bx}+\mathrm{C}$ put $\mathrm{x}=0 \quad \mathrm{C}=1$ Put $\mathrm{x}=1$ $3=\mathrm{A}+\mathrm{B}+1$ $A+B=2$ .........(i) put $x=-1$ $-1=A-B+1$ $A-B=-2$ .........(ii) (i) $+(i i)$ $A=0$ $B=2$

Hence, the correct answer is option: 'C'.

More Quadratic Equation Questions