Question

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) q(x) + r(x) where deg r(x) = 0

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

Answer 1

The correct option is C $f\left(x\right)=x^5-4x^3+x^2+3x+1,g\left(x\right)=x^3-3x+1,q\left(x\right)=x^2-1,r\left(x\right)=2$

Since in all the four options; $g\left(x\right),q\left(x\right),r\left(x\right)$ are the same, we just multiply $g\left(x\right)$ and $q\left(x\right)$ and add $r\left(x\right)$ to get $f\left(x\right)$ as the answer.
Thus, $f\left(x\right)=(x^3-3x+1)\times (x^2-1)+2$
$\Rightarrow f\left(x\right)=x^2(x^3-3x+1)-1(x^3-3x+1)+2$
i.e. $f\left(x\right)=x^5-3x^3+x^2-x^3+3x-1+2$
i.e. $f\left(x\right)=x^5-4x^3+x^2+3x+1$