Exercise 5.3.1

Proof. Write

(We suppose $a_i=0$ if $i>m$ or $i<0$, $b_j=0$ if $j>m$ or $j<0$.)

(a)

Write $N=\max <!--\; FUNCTION\; APPLICATION\; -->(n,m)$ : then $$g=\underset{i=0}{\overset{N}{\sum }}{a}_i{x}^i,h=\underset{i=0}{\overset{N}{\sum }}{b}_i{x}^i.$$

$$\begin{array}{lll}\hfill g^{\prime }=\underset{i=1}{\overset{N}{\sum }}i{a}_i{x}^{i-1},h^{\prime }=\underset{i=1}{\overset{N}{\sum }}i{b}_i{x}^{i-1}.& & \hfill \end{array}$$

If $a,b\in F$, then

$$a{g}^{\prime }+b{h}^{\prime }=\underset{i=1}{\overset{N}{\sum }}i(a{a}_i+b{b}_i)x^{i-1}.$$

Moreover

$$\begin{array}{llll}\hfill \mathit{ag}+\mathit{bh}& =\underset{i=0}{\overset{N}{\sum }}(a{a}_i+b{b}_i)x^i,& \hfill & \\ \hfill {(\mathit{ag}+\mathit{bh})}^{\prime }& =\underset{i=1}{\overset{N}{\sum }}i(a{a}_i+b{b}_i)x^{i-1},& \hfill & \end{array}$$

thus

$$\begin{array}{lll}\hfill {(\mathit{ag}+\mathit{bh})}^{\prime }=a{g}^{\prime }+b{h}^{\prime }.& & \hfill \end{array}$$

(b)

By (1), the definition of the product of two polynomials gives $$\mathit{gh}=\underset{k=0}{\overset{m+n}{\sum }}\left(\underset{i=0}{\overset{k}{\sum }}{a}_i{b}_{k-i}\right)x^k.$$

Thus

$${(\mathit{gh})}^{\prime }=\underset{k=1}{\overset{m+n}{\sum }}k\left(\underset{i=0}{\overset{k}{\sum }}{a}_i{b}_{k-i}\right)x^{k-1}=\underset{k=0}{\overset{m+n-1}{\sum }}(k+1)\left(\underset{i=0}{\overset{k+1}{\sum }}{a}_i{b}_{k+1-i}\right)x^k.$$

As

$$\begin{array}{llll}\hfill g^{\prime }& =\underset{i=1}{\overset{n}{\sum }}i{a}_i{x}^{i-1}=\underset{i=0}{\overset{n-1}{\sum }}(i+1)a_{i+1}{x}^i,& \hfill & \\ \hfill h^{\prime }& =\underset{j=1}{\overset{m}{\sum }}j{b}_j{x}^{j-1}=\underset{j=0}{\overset{m-1}{\sum }}(j+1)b_{j+1}{x}^j,& \hfill & \end{array}$$

we obtain

$$\begin{array}{llll}\hfill g^{\prime }h& =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k}{\sum }}(i+1)a_{i+1}{b}_{k-i}\right)x^k& \hfill & \\ \hfill & =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=1}{\overset{k+1}{\sum }}i{a}_i{b}_{k+1-i}\right)x^k& \hfill & \\ \hfill & =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k+1}{\sum }}i{a}_i{b}_{k+1-i}\right)x^k,& \hfill & \\ \hfill & & \hfill \end{array}$$

and also

$$\begin{array}{llll}\hfill g{h}^{\prime }& =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k}{\sum }}(k+1-i)a_i{b}_{k+1-i}\right)x^k& \hfill & \\ \hfill & =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k+1}{\sum }}(k+1-i)a_i{b}_{k+1-i}\right)x^k.& \hfill & \end{array}$$

Thus

$$\begin{array}{llll}\hfill g^{\prime }h+g{h}^{\prime }& =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k+1}{\sum }}i{a}_i{b}_{k+1-i}+\underset{i=0}{\overset{k+1}{\sum }}(k+1-i)a_i{b}_{k+1-i}\right)x^k& \hfill & \\ \hfill & =\underset{k=0}{\overset{m+n-1}{\sum }}\left(\underset{i=0}{\overset{k}{\sum }}(i+k+1-i)a_i{b}_{k+1-i}\right)x^k& \hfill & \\ \hfill & =\underset{k=0}{\overset{m+n-1}{\sum }}(k+1)\left(\underset{i=0}{\overset{k+1}{\sum }}{a}_i{b}_{k+1-i}\right)x^k& \hfill & \\ \hfill & ={(\mathit{gh})}^{\prime }.& \hfill & \end{array}$$

We have proved the equations 5.6 :

$$\begin{array}{llll}\hfill {(\mathit{ag}+\mathit{bh})}^{\prime }& =a{g}^{\prime }+b{h}^{\prime },& \hfill & \\ \hfill {(\mathit{gh})}^{\prime }& =g^{\prime }h+g{h}^{\prime }.& \hfill & \end{array}$$