Question

If $f\left(x\right),g\left(x\right)\in R\left[x\right]$ and $f\left(x\right)+g\left(x\right)\ne 0_R$, show that

$deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{degf\left(x\right),deg\left[g\left(x\right)\right]\right\}$

Short answer

It is proved that$deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{degf\left(x\right),deg\left[g\left(x\right)\right]\right\}$

Step-by-step solution

Polynomial Arithmetic: 

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function f(x)+g(x) exists.

Polynomial Operations

Let the given functions be:

$$\begin{gathered} f\left(x\right)=\sum _{i=0}^m{a}_i{x}^i.......\left\{m=degf\left(x\right)\right\} \\ g\left(x\right)=\sum _{j=0}^m{b}_j{x}^j.......\left\{n=degg\left(x\right)\right\} \end{gathered}$$

Now, assuming:

$$\begin{gathered} i>m\Rightarrow a_i=0 \\ j>n\Rightarrow b_j=0 \end{gathered}$$

We have,

$\begin{array}{rcl}f\left(x\right)+g\left(x\right)& =& \left\{\sum _{i=0}^m{a}_i{x}^i\right\}+\left\{\sum _{j=0}^m{b}_j{x}^j\right\}\\ & =& \sum _{i=0}^{max\left(m,n\right)}\left(a_i+b_i\right)x^i\end{array}$

This implies that the degree of above expression will lie within:

$0\le i\le max\left(m,n\right).......\left\{a_i+b_i\ne 0\right\}$

.

This results: $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$

Hence proved, $deg\left[f\left(x\right)+g\left(x\right)\right]\le max\left\{deg\left[f\left(x\right)\right],deg\left[g\left(x\right)\right]\right\}$.