Introduction
We know that an expression involving variables, constants, and operators (+,−,×,÷) is called an algebraic expression.
Examples: ${x}^{2},5x+y,2x+\frac{1}{x}+{x}^{3}$
Polynomial: An algebraic expression in which the power of variables is a nonnegative integer is called a polynomial.
${x}^{3}+{x}^{0}$ is a polynomial, but ${x}^{2}+5$ and $\sqrt{x}+2{x}^{2}+6$ is not a polynomial.
Polynomials in One Variable
Consider a square with sides a. The area of the square is a^{2}. This is an algebraic expression.
Similarly, we are familiar with many algebraic expressions such as $5a,{a}^{3}+2a+6,9{a}^{2}+4a,\sqrt{2a}+\sqrt{5}$. Such expressions are called polynomials in one variable. Here the variable is ‘a’
Generally, polynomials in one variable are the polynomials of the form $a{x}^{m}+b$ where a and b are constants, x is a variable, and m is a positive integer. A polynomial can be denoted as p(x) or q(x) or f(x) etc. if it contains the variable x
Terms of a polynomial: A term of a polynomial is of the form $a{x}^{m}\phantom{\rule{0ex}{0ex}}$ where a is a constant, x is a variable, and m is any positive integer. A polynomial can have only a finite number of terms.
Examples:
1) In the polynomial $\sqrt{2{r}^{2}}+25r$, there are two terms $\sqrt{2{r}^{2}}and25r$
2) The polynomial $5{x}^{2}+4x+2$ has three terms: $5{x}^{2},4xand2$
Coefficient of a term: It is the constant attached to a term in a polynomial.
Example: In the polynomial $3{x}^{2}17$, the coefficient of ${x}^{2}$ is 3 and the coefficient of ${x}^{0}$ is 17
Constant polynomial: This is a polynomial without any variable, i.e., a polynomial has only one term, which is a constant.
Examples: $2,35,\sqrt{26}$ etc.
Zero polynomial: The constant polynomial 0 is known as the zero polynomial.
Degree of a polynomial: The highest power of the variable in a polynomial is called the degree of the polynomial.
Examples:

Consider the polynomial $f\left(x\right)=2{x}^{6}3{x}^{4}+x+5$ The term with the highest power of x is $2{x}^{6}$. The exponent of x in this term is 6. So, the degree of the polynomial is 6.

Similarly, in the polynomial $f\left(y\right)=3{y}^{5}8{y}^{2}6$, the term with the highest power of y is $3{y}^{5}$. The exponent of y in this term is 5, and thus the degree of the polynomial is 5
Note

The degree of a nonzero constant polynomial is zero.

The degree of a zero polynomial is not defined.
Example 1
Find the degree of the polynomials $5{x}^{5}+4x+2$ and $3{x}^{3}+2x+5$
Solution:

$5{x}^{5}+4x+2$
The highest power of the variable is 5, and hence the degree of the polynomial is 5
2. $3{x}^{3}+2x+5$
The highest power of the variable is 3, and hence the degree of the polynomial is 3
Example 2
Consider the given polynomial p(x) and answer the following questions:
$p\left(x\right)=3{x}^{7}+\sqrt{3{x}^{3}}3{x}^{2}+19$

Find the number of terms in p(x).

Write the coefficients of eq and eq in p(x).

Find the degree of p(x).
Solution:

There are 4 terms in p(x): $3{x}^{7},\sqrt{3{x}^{3}},3{x}^{2}$ and 19

The coefficient of ${x}^{0}$ 19, and the coefficient of ${x}^{2}$ is 3

The degree of p(x) is 7 since the highest power of the variable x is 7
Classification of polynomials based on the number of terms
A term of a polynomial is of the form ax^{n} where a is a constant, x is a variable, and n is any nonnegative integer. A polynomial can have a finite number of terms.
The given figure shows the classification of polynomials based on the number of terms.
Examples:
Classify the following polynomials as monomial, binomial, or trinomial
$1.{x}^{2}+{y}^{2}\phantom{\rule{0ex}{0ex}}2.2y3y+4{y}^{3}\phantom{\rule{0ex}{0ex}}3.{p}^{2}q+p{q}^{2}\phantom{\rule{0ex}{0ex}}4.9pqr$
Solution:
$1.{x}^{2}+{y}^{2}:Binomial\phantom{\rule{0ex}{0ex}}2.2y3y+4{y}^{3}:Trinomial\phantom{\rule{0ex}{0ex}}3.{p}^{2}q+p{q}^{2}:Binomial\phantom{\rule{0ex}{0ex}}4.9pqr:Monomial$
Classification of polynomials based on the degree of polynomials
The polynomial of degree n is an expression of the form
${a}_{n}{x}^{n}+{a}_{n1}{x}^{n1}+.......+{a}_{1}x+{a}_{0}$ where ${a}_{0},{a}_{1},{a}_{2}.....{a}_{n}$ are constants and ${a}_{n}\ne 0$
If all the constants ${a}_{0},{a}_{1},{a}_{2}........{a}_{n}=0$, a zero polynomial is obtained, which is denoted by 0. The degree of a zero polynomial is not defined.
The given figure shows the classification of polynomials based on the degree of polynomials:
Examples:
$\left(a\right)p\left(x\right)={x}^{2}+2x19\phantom{\rule{0ex}{0ex}}\left(b\right)q\left(x\right)={x}^{3}3x\phantom{\rule{0ex}{0ex}}\left(c\right)r\left(x\right)=x+3\phantom{\rule{0ex}{0ex}}\left(d\right)s\left(x\right)={x}^{3}\sqrt{5x}22$
Classify the following polynomials as linear, quadratic, and cubic polynomials
Solution:
Linear Polynomial : $r\left(x\right)=x+3$
Quadratic Polynomial: $p\left(x\right)={x}^{2}+2x19$
Cubic Polynomial: $q\left(x\right)={x}^{3}3x\phantom{\rule{0ex}{0ex}}s\left(x\right)={x}^{3}\sqrt{5x}22$