Is this a polynomial? $$x^4 + x^3 + x + 1$$
Notice how I skipped $x^2$. Do "polynomials" need to have a sequence of exponents that start from $1$ and go up by $1$ and only $1$ each time? Thanks
 A: A polynomial is any sum of the type
$$a_0 + a_1x + a_2x^2 + \cdots + a_n x^n.$$
By taking $n=4$ and setting the coefficients at $a_0=1, a_1=1, a_2=0, a_3=1, a_4=1$, what do you get?
A: That is indeed a polynomial!  You do not need the exponents increasing by $1$.  For example, $x^{100} + x^2 + 2$ is a polynomial.  By definition, any function $p(x)$ is a polynomial if it can be written in the form:
$$p(x) = \sum_{k=0}^n c_kx^k = c_0 + c_1x + c_2x^2 + ... + c_nx^n$$
Where the $c_k$'s are arbitrary elements of a particular ring.  If you haven't seen rings before, there's no need to get caught up in too much theory.  Outside of abstract algebra, the ring is usually the integers, the rational numbers, the real numbers, or the complex numbers.  The specific reason why the exponents do not need to increase by $1$ each time is because $0$ is an element of all rings ($0$ is an integer, rational, real, and complex number).  Thus, $c_k$ can be $0$ for any $k$.
Even if $c_k$ is $0$ for all $k$, then $p(x)$ is still a polynomial!  Algebraists often call $p(x) = 0$ the "zero polynomial".

There is one last restriction:  For $p(x)$ to be a polynomial, the $n$ in $\displaystyle \sum_{k = 0}^n c_kx^k$ must be finite.  Otherwise, $\displaystyle \sum_{k=0}^\infty c_kx^k$ would be called a power series.
A: Yes that is a polynomial. Any integer is a polynomial. Any combination of multiplcation/addition of numbers and variables will be a polynomial.
