Question

Polynomials $p\left(x\right),g\left(x\right),q\left(x\right)$ and $r\left(x\right)$, which satisfy the division algorithm and "deg $p\left(x\right)=$ deg $q\left(x\right)$" are

• A. $p\left(x\right)=2x^2+2x+8,g\left(x\right)=4x+1$;
  $q\left(x\right)=x^2;r\left(x\right)=1$
  
• B. $p\left(x\right)=2x^2+2x+8,g\left(x\right)=5$;
  $q\left(x\right)=4;r\left(x\right)=4x-1$
  
• C. $p\left(x\right)=2x^2+2x+8,g\left(x\right)=2$;
  $q\left(x\right)=x^2+x+4;r\left(x\right)=0$
  
• D. $p\left(x\right)=x^2+x+3,g\left(x\right)=2x+3$;
  $q\left(x\right)=2x^2+x;r\left(x\right)=3x-2$
  

Answer 1

Answer: (C)

$p\left(x\right)=2x^2+2x+8,g\left(x\right)=2$;

$q\left(x\right)=x^2+x+4;r\left(x\right)=0$
degree of $p\left(x\right)$ and $q\left(x\right)$ are equal in (A),(C),(D)

according to division algorithm, $p\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

in option (C), $g\left(x\right)q\left(x\right)=2(x^2+x+4)=2x^2+2x+8+0=g\left(x\right)q\left(x\right)+r\left(x\right)=p\left(x\right)$

hence option (C) is correct answer.