Question

Let R(x) be the remainder when the polynomial $x^{401}=x^{301}+x^{201}+x^{101}+x$ is divided by $x^4-x$ Then which of the following statements is/are correct?

• A. R(-1) equals -1
• B. $R\left(2\right)R\left(\frac{1}{2}\right)$ equals $\frac{65}{2}$
• C. R(–2) equals (–4)
• D. R(1) equals 0

Answer 1

Answer: (A), (B), (C)

The correct options are
A R(-1) equals -1
B $R\left(2\right)R\left(\frac{1}{2}\right)$ equals $\frac{65}{2}$
C R(–2) equals (–4)

given $x^{401}+x^{301}+x^{201}+x^{101}+x=(x-1)x(x-\omega )(x-{\omega }^2)$ given
$R\left(x\right)=a{x}^3+b{x}^2+cx+d$
out $x=d,x=1,n=\omega ,n={\omega }^{2}givend=0,a=1,b=c=2R\left(x\right)=x^3+2x^2+2x$