Find the remainder when $x^{3}+3 x^{2}+3 x+1$ is divided by <br/> <br/>(i) $x+1$ <br/> <br/>(ii) $x-\frac{1}{2}$ <br/> <br/>(iii) $x$(iv) $x+\pi$ <br/> <br/>(v) $5+2 x$

Solution:

(i) $x+1$

By long division,



Therefore, the remainder is 0 .

(ii) $x-\frac{1}{2}$

By long division,



Therefore, the remainder is $\frac{27}{8}$.

(iii) $X$

By long division,



Therefore, the remainder is 1 .

(iv) $x+\pi$

By long division,



Therefore, the remainder is $-\pi^{3}+3 \pi^{2}-3 \pi+1$.

(v) $5+2 x$

By long division,



Therefore, the remainder is $-\frac{27}{8}$