Suggested languages for you:

Americas

Europe

12

Expert-verifiedFound in: Page 94

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**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 **

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

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

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.

Let the given functions be:

$f\left(x\right)=\sum _{i=0}^{m}{a}_{i}{x}^{i}.......\left\{m=degf\left(x\right)\right\}\phantom{\rule{0ex}{0ex}}g\left(x\right)=\sum _{j=0}^{m}{b}_{j}{x}^{j}.......\left\{n=degg\left(x\right)\right\}$

Now, assuming:

$i>m\Rightarrow {a}_{i}=0\phantom{\rule{0ex}{0ex}}j>n\Rightarrow {b}_{j}=0$

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

94% of StudySmarter users get better grades.

Sign up for free