Question

(a) If $\mathbf{(}{\mathbf{a}}_{\mathbf{1}}\mathbf{,}{\mathbf{a}}_{\mathbf{2}}\mathbf{,}\mathbf{...}\mathbf{)}$ and$\mathbf{(}{\mathbf{b}}_{\mathbf{1}}\mathbf{,}{\mathbf{b}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{..}\mathbf{)}$ are polynomials, show that their sum is a polynomial (that is, after some point all coordinates of the sum are zero).

(b) Show thatrole="math" localid="1658910018438" $\mathbf{(}{\mathbf{a}}_{\mathbf{1}}\mathbf{,}{\mathbf{a}}_{\mathbf{2}}\mathbf{,}\mathbf{...}\mathbf{)}\mathbf{\odot }\mathbf{(}{\mathbf{b}}_{\mathbf{1}}\mathbf{,}{\mathbf{b}}_{\mathbf{2}}\mathbf{,}\mathbf{...}\mathbf{)}$ is a polynomial.

Short answer

1. The addition of given two polynomial is $\mathrm{P}({\mathrm{x}}_1+{\mathrm{x}}_2)=({\mathrm{a}}_1+{\mathrm{b}}_1)+({\mathrm{a}}_2+{\mathrm{b}}_2)\mathrm{x}$.
2. It is shown that $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})\odot ({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})$is a polynomial.

Step-by-step solution

Introduction of polynomial 

Consider the polynomial in $\mathit{x}$and $\mathbf{ℝ}$will be express as;

role="math" localid="1658910328362" $\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{a}}_{\mathbf{0}}{\mathbf{x}}^{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{x}}^{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{..}\mathbf{+}{\mathbf{a}}_{\mathbf{n}}{\mathbf{x}}^{\mathbf{n}}$

Polynomial is always finite but sequence can be finite and infinite.

Show that the sum of (a1, a2,...) and (b1, b2,...) is a polynomial 

(a)

Consider the sequences $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})$and $({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})$are the polynomials then the polynomial for the sequence $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})$is,

$\begin{array}{c}\mathrm{P}\left({\mathrm{x}}_1\right)={\mathrm{a}}_1{\mathrm{x}}^0+{\mathrm{a}}_2{\mathrm{x}}^1+0\cdot {\mathrm{x}}^2+0\cdot {\mathrm{x}}^3+0\cdot {\mathrm{x}}^4\\ ={\mathrm{a}}_1+{\mathrm{a}}_2\mathrm{x}\end{array}$

For the sequence,$(b_1,b_2,\mathrm{...})$ the polynomials is,

$\begin{array}{c}P\left(x_2\right)=b_1{x}^0+b_2{x}^1+0\cdot x^2+0\cdot x^3+0\cdot x^4\\ =b_1+b_{2}x\end{array}$

Add the above two sequences.

$({\mathrm{a}}_1,{\mathrm{a}}_2,0,0,0,\mathrm{...})+({\mathrm{b}}_1,{\mathrm{b}}_2,0,0,0,\mathrm{...})=({\mathrm{a}}_1+{\mathrm{b}}_1,{\mathrm{a}}_2+{\mathrm{b}}_2,0,0,0,\mathrm{...})$

The polynomial for the sequence$({\mathrm{a}}_1+{\mathrm{b}}_1,{\mathrm{a}}_2+{\mathrm{b}}_2,0,0,0,\mathrm{...})$ is,

role="math" localid="1658910585116"

Therefore, the addition of given two polynomial is$\mathrm{P}({\mathrm{x}}_1+{\mathrm{x}}_2)=({\mathrm{a}}_1+{\mathrm{b}}_1)+({\mathrm{a}}_2+{\mathrm{b}}_2)\mathrm{x}$ .

Show that (a1, a2,...)⊙(b1, b2,...) is a polynomial

(b)

Consider the sequences $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})$and $({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})$, these two sequence is infinite sequences of a finite coefficients of a polynomial.

So, the rest entries of the polynomial will be zeroes because polynomial will never be infinite.

Therefore, the sequence will be as follows.

$({\mathrm{a}}_1,{\mathrm{a}}_2,0,0,0,\mathrm{...})$and $({\mathrm{b}}_1,{\mathrm{b}}_2,0,0,0,\mathrm{...})$,

Evaluate $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})\odot ({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})$.

$({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})\odot ({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})=({\mathrm{a}}_1\cdot {\mathrm{b}}_1,{\mathrm{a}}_1\cdot {\mathrm{b}}_2,{\mathrm{a}}_2\cdot {\mathrm{b}}_1,{\mathrm{a}}_2\cdot {\mathrm{b}}_2,0,0,0,\mathrm{...})$

The polynomial for the sequencerole="math" localid="1658910793522" $(a_1\cdot b_1,a_1\cdot b_2,a_2\cdot b_1,a_2\cdot b_2,0,0,0,\mathrm{...})$ will be as follows.

Therefore, $({\mathrm{a}}_1,{\mathrm{a}}_2,\mathrm{...})\odot ({\mathrm{b}}_1,{\mathrm{b}}_2,\mathrm{...})$is a polynomial.