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 $q\left(x\right)=$ deg $r\left(x\right)$", are

• A. $p\left(x\right)=2x^2+x;g\left(x\right)=2x^2-4$;
  $q\left(x\right)=2x-7;r\left(x\right)=-x+2$
• B. $p\left(x\right)=x^2+x-3;g\left(x\right)=x^2+x-1$;
  $q\left(x\right)=7;r\left(x\right)=-5$
• C. $p\left(x\right)=x^2+x;g\left(x\right)=x^2-4$;
  $q\left(x\right)=2x-1;r\left(x\right)=-x-2$
  
• D. $p\left(x\right)=2x^2+2x+8;g\left(x\right)=x^2+x+9$;
  $q\left(x\right)=2;r\left(x\right)=-10$

Answer 1

Answer: (D)

$p\left(x\right)=2x^2+2x+8;g\left(x\right)=x^2+x+9$;

$q\left(x\right)=2;r\left(x\right)=-10$
according to division algorithm $p\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

degree of $q\left(x\right)$ is equal to $r\left(x\right)$ in all options.

only (D) option satisfies $p\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$

$g\left(x\right)q\left(x\right)=2(x^2+x+9)=2x^2+2x+18=p\left(x\right)+10=p\left(x\right)-r\left(x\right)$ hence $p\left(x\right)=q\left(x\right)g\left(x\right)+r\left(x\right)$ in (D) satisfies division algorithm