Question

Let$f\left(x\right),g\left(x\right),h\left(x\right)\in F\left[x\right]$ and $r\in F$.

(a) If$f\left(x\right)=g\left(x\right)+h\left(x\right)$ in $F\left[x\right]$, show that $f\text{'}\left(r\right)=g\left(r\right)+h\left(r\right)$in$F$ .

(b) If $f\left(x\right)=g\left(x\right)h\left(x\right)$in $F\left[x\right]$, show that $f\left(r\right)=g\left(r\right)h\left(r\right)$in $F$.

Short answer

(a) and (b) both are proved.

Step-by-step solution

Determine f'r=gr+hr in F

Consider that, $f\left(x\right)={\sum }_{k=0}^m{f}_k{x}^k,g\left(x\right)={\sum }_{i=0}^m{g}_i{x}^i$and $h\left(x\right)={\sum }_{j=0}^m{h}_j{x}^j$, here, assume$f_m\ne 0$ but allow $g_m=0or{h}_m=0$.

$f\left(x\right)=g\left(x\right)+h\left(x\right)$it means that $f_k=g_k+h_k$for all $0\le k\le m$.

$\begin{array}{c}f\left(r\right)=\sum _{k=0}^m{f}_k{r}^k\\ =\sum _{k=0}^m\left(g_k+h_k\right)r^k\\ =\left(\sum _{i=0}^m{g}_i{r}^i\right)+\left(\sum _{j=0}^m{h}_j{r}^j\right)\\ =g\left(r\right)+h\left(r\right)\end{array}$

Determine fr=gr+hr in F

Similarly,$f\left(x\right)=g\left(x\right)h\left(x\right)$ that means$f_k={\sum }_{i+j=k}{g}_i{h}_j$ such that,

$\begin{array}{c}f\left(r\right)=\sum _{k=0}^m{f}_k{r}^k\\ =\sum _{k=0}^m\left(\sum _{i+j=k}^m{h}_j{r}^j\right)\\ =g\left(r\right)h\left(r\right)\end{array}$

Therefore, it can be concluded,$f\left(x\right)=\left(x-a\right)h\left(x\right)+cthatf\left(a\right)=c$