Question

Give example of polynomialsp(x),g(x),q(x) and r(x), which satisfies the division algorithm and
i) deg p(x)=deg q(x)
ii)deg q(x)=deg r(x)
iii)deg r(x)=0

Answer 1

We can write many such examples
$Degp\left(x\right)=degq\left(x\right)$
We can write many such examples
$p\left(x\right)=x^2,q\left(x\right)=x^{2}g\left(x\right)=1R\left(x\right)=0$

(ii) $degq\left(x\right)=degr\left(x\right)$
$Q\left(x\right)=x,R\left(x\right)=x,p\left(x\right)=x^3+xg\left(x\right)=x^2$

(iii) $degr\left(x\right)=0$
$Q\left(x\right)=1,R\left(x\right)=1,p\left(x\right)=x+1,g\left(x\right)=x$