When we have two polynomials written in this form $\sum_{i=0}^{n} a_{i}x^{i} + \sum_{j=0}^{m} b_{j}x^{j}$, how can their sum be equal to $\sum_{k=0}^{MAX(n,m)} (a_{k}+b_{k})x^{k}$?
If we take for example $n=3$ and $m=2$ we get:
$\sum_{i=0}^{3} a_{i}x^{i}= a_{0}+ a_{1}x + a_{2}x^{2} + a_{3}x^{3}$
$\sum_{j=0}^{2} b_{j}x^{j}= b_{0}+ b_{1}x + b_{2}x^{2}$
When we add first and second sum:
$(a_{0}+ a_{1}x + a_{2}x^{2} + a_{3}x^{3}) + (b_{0}+ b_{1}x + b_{2}x^{2})$
we get:
$a_{0}+b_{0} + x(a_{1}+b_{1}) + x^{2}(a_{2}+b_{2})+a_{3}x^{3}$
How can that be same as $\sum_{k=0}^{3} (a_{k}+b_{k})x^{k}$ ( if our max is 3)? when this sum is equal to :
$\sum_{k=0}^{3} (a_{k}+b_{k})x^{k}=a_{0}+b_{0} + x(a_{1}+b_{1}) + x^{2}(a_{2}+b_{2})+x^{3}(a_{3}+b_{3})$?
