When we multiply a number (coefficient) for an unknown (variable) is a monomial. But what if we add instead of multiply?
$$$x^6+10$$$ $$$x+1$$$
What happens when we add monomials that are similar? and if we subtract them?
When we join not similar monomials by adding or subtracting them we get a polynomial.
$$$2x^2+x1$$$ that is the result of adding the monomials $$2x^2$$ and $$x$$, and subtracting the monomial $$1$$.
Or $$$3x^5x^2+x5$$$ that is the result of adding the monomials $$3x^5$$ and $$x$$, and subtracting the monomials $$x^2$$ and $$5$$.
In mathematics, to call polynomials we use one letter followed by a parenthesis with the variable (or variables, separated by commas). So the above examples would be:
$$p(x)=2x^2+x1$$ and $$q(x)=3x^5x^2+x5$$
If there is more than one variable, as we said:
$$p(x,y)=x^6y+xyx$$
$$q(x,y,z)=xyz^2+xyzxy^3zzyz+zyz$$
$$r(x,y,z,t)=xyzt$$
Be careful in the way we represent polynomials because it is easy to make notation mistakes.
$$q(x,y)=3x^2y+4x$$, $$q(x)=3x^2y+4x$$
In the first polynomial, "$$y$$" acts as a variable. However, in the second, the "$$y$$" is a coefficient (which value is $$y$$, a number that we don't know a priori).
So they are two different polynomials (For example, the first one has degree $$3$$ and the second one has degree $$2$$).
Now, using as an example the polynomial $$p(x)=2x^2+x1$$, we define the following characteristics of a polynomial:

Variable/s of the polynomial: unknown or unknowns that we find in the polynomial. In the polynomial $$p(x), x$$.

Degree of the polynomial: the greatest exponent of all monomials which has the polynomial. In our example $$max\{2,1,0\}=2$$

Leading coefficient: the coefficient of the monomial which has the higher degree. In our case, $$2$$.
 Independent term: the coefficient of the monomial with exponent zero. If there is no such monomial then is equal to $$0$$. In our case, it is $$1$$.
Classification of polynomials
We can categorize the polynomials according to their characteristics.
Classification of polynomials according to their degree
 Degree zero: They are coefficients. $$$q(x)=1$$$ $$$q(x)=\dfrac{1}{2}$$$
 The first degree: $$$q(x)=x1$$$ $$$q(y)=3y\dfrac{3}{4}$$$ $$$p(y)=\dfrac{y}{2}+\dfrac{1}{4}$$$
 The second degree: $$$p(z)=z^2+3z9$$$ $$$p(x)=\dfrac{x^2}{3}+2x$$$ $$$q(z)=z^2\dfrac{10}{3}$$$
 Third degree: $$$r(t)=t^3+t^2+1$$$ $$$p(t)=\dfrac{t^3}{4}+\dfrac{t^2}{2}t+10$$$ $$$q(x)=x^3\dfrac{1}{4}$$$
And, in this way, we might continue to the number that we need.
Classification of polynomials according to their coefficients
 Finished polynomial: it has all the coefficients other than zero. $$$p(x)=x^3+x^2+x+1$$$ $$$p(x,y)=2x^2+y^2xy+x+y\dfrac{1}{3}$$$ $$$r(t)=t^24t+9$$$
 Incomplete polynomial: it has some coefficient which value is zero. $$$p(x)=x^3+x+1$$$ $$$p(x,y)=2x^2+y^2+x+y\dfrac{1}{3}$$$ $$$r(t)=t^24t$$$
 Null polynomial: all its coefficient are equal to zero. $$$p(x)=0$$$
Classification of polynomials according to the degrees of their monomials
 Ordered polynomial: the monomials are written from greater to lesser degree. $$$p(x)=x^4+x^3+x^2+x+1$$$ $$$q(x)=x^6+x^4+x^2+x+1$$$ $$$r(x)=x^{100}+x^2+2x$$$
 Homogeneous polynomial: all their monomials have the same degree. $$$p(x)=2x$$$ $$$p(x,y)=3x^2y+4x^3+2xy^2$$$ $$$p(x,y)=\dfrac{xy}{2}+x^2+y^2$$$
 Heterogeneous polynomial: not all their monomials have the same degree. $$$p(x)=2x1$$$ $$$p(x,y)=3x^2y+4x^3+2xy^2$$$ $$$p(x,y)=\dfrac{xy}{2}+x^2y+y^2$$$
 Equal polynomials. They are those which satisfy the next conditions:
 They have the same degree.
 The coefficients of the monomials of the same degree are equal. $$$p(x)=3x^2+1$$$ $$$q(x)=1+3x^2$$$ $$$p(x,y)=xy+4x1$$$ $$$q(y,x)=1+4x+yx$$$