How does knowing the degree of a polynomial help us? Any polynomial such as $5x^6+2x^3+8$ has the degree of polynomial 6 but not 3 as it is said that the highest power of any term in a polynomial is the degree of that polynomial. But why? And how knowing the degrees of the polynomial will be useful to us?
 A: To your question of the usefulness of the concept of degree for polynomials, here’s one possible answer:
The polynomials of degree zero are the constants: boringly simple. Those of degree one are the linear functions: not quite boring, but still simple. Those of degree two are those whose graphs are parabolas: now we’re getting some interesting behavior. And so it goes.
That was an informal answer. A more formal answer is that if $f$ is a polynomial, we are often interested in the roots of $f$, namely the numbers $\lambda$ such that $f(\lambda)=0$. There is a Theorem, not so difficult, that a polynomial of degree $n$, with real coefficients, has at most $n$ real roots. You can see that $x^2+1$ has no real roots, though, so the story is perhaps more complicated than we’d like. And $x^2$, also of degree two, has just one root, namely $\lambda=0$.
Now the lovely fact, one of those things that makes mathematics esthetically satisfying, is that a real polynomial of degree $n$ (or a complex one, for that matter) has precisely $n$ roots in the complex domain, “if we count multiplicity”. The way to do that I’ll leave for you to discover in your mathematical future, but it’s something you can look forward to.
A: Here's an analogy: writing an integer in decimal notation is like writing out a polynomial. I hope you are familiar with decimal notation; for example,
$$
1045 = 1\cdot10^3+0\cdot10^2+4\cdot10^1+5\cdot1.
$$
In this analogy, the number of digits of an integer in decimal representation, minus one, corresponds to the degree of a polynomial.
Now the degree of the sum of two polynomials is the larger degree of the two for the same reason that when you add two integers – at least in the case when no carrying occurs (this part of the analogy doesn't carry over well) – the number of digits of the result will be the greater of the number of digits of the two integers you began with. For example, $1000$ has four digits and $45$ has two digits so $1045$ has $4$ digits.
A: If you take a polynomial of degree $x$, and you add it to a polynomial of degree $y$, where $x>y$, the degree of the resulting polynomial is $x$. This is because when you add polynomials, you combine like terms, and the final polynomial includes the term with degree $x$, which is the biggest power. Therefore, every time you add polynomials and check the degree, it must be the bigger degree.
