Question.
Give examples of polynomials p(x), g(x),q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0.
Give examples of polynomials p(x), g(x),q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0.
Solution:
(i) $p(x)=2 x^{2}+2 x+8, g(x)=2 x^{0}=2$
$q(x)=x^{2}+x+4 ; r(x)=0$
(ii) $\mathrm{p}(\mathrm{x})=2 \mathrm{x}^{2}+2 \mathrm{x}+8 ; \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{x}+9$;
$q(x)=2 ; r(x)=-10$
(iii) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}+5 ; \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+1 ;$
$\mathrm{q}(\mathrm{x})=\mathrm{x} ; \mathrm{r}(\mathrm{x})=5$
(i) $p(x)=2 x^{2}+2 x+8, g(x)=2 x^{0}=2$
$q(x)=x^{2}+x+4 ; r(x)=0$
(ii) $\mathrm{p}(\mathrm{x})=2 \mathrm{x}^{2}+2 \mathrm{x}+8 ; \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{x}+9$;
$q(x)=2 ; r(x)=-10$
(iii) $\mathrm{p}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}+5 ; \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+1 ;$
$\mathrm{q}(\mathrm{x})=\mathrm{x} ; \mathrm{r}(\mathrm{x})=5$