What we discuss here:
 Meaning of the terms of an expression
 What is a polynomial and what is not a polynomial
 Degree of a polynomial
 Like and unlike terms
 How to add, subtract, and multiply polynomials
Terms of an expression
The parts of an expression connected by the signs + or – are called the terms of the expression. For example, $x^{2}$, $2xy$, and $\sqrt{x}$ are the terms of the expression $$x^{2}+2xy+\sqrt{x},$$ and the expression $$2aab+c^{2}$$ is made up of three terms: $2a$,$ab$, and $c^{2}$.
Polynomials
A polynomial or more precisely a polynomial in $x$ is an algebraic expression consisting of terms in the form $$ax^{k}$$ where $k$ is a nonnegative integers (that is zero or a natural number 0, 1, 2, …) and $a$ is a real number called the coefficient of the term. For example,
\[3,\quad x,\quad7x^{51},\quadx^{3}+\frac{1}{\sqrt{14}}x3,\quad\text{ and}\quad 7x^{5}\pi x^{4}\sqrt{2}x^{3}x^{2}+x3\]
are all polynomials. In the last example, the coefficients are $7,\pi,\sqrt{2},1, 1,$ and $3$.
 Note that any constant is also a polynomial because it can be written as $ax^{0}$; for example $3=3x^{0}$.
 Of course, instead of a polynomial in $x$, we may have polynomials in $y$ or $z$ or any other letter. For example, $4z^{3}0.5z+1$ is a polynomial in $z$.
Monomials, binomials, and Trinomials
 A polynomial that has only one term is called a monomial. For example, $4x^{3}$ and $\sqrt{7}x^{2}$ are two monomials.
 A polynomial that has two terms is called a binomial. For example, $3x+2$ and $4x^3+x$ are two binomials.
 A polynomial that has three terms is called a trinomial. For example, $3x+8x^59$ is a trinomial.
Degree of a Polynomial
The largest exponent in a polynomial is called the degree of the polynomial. In $$7x^{5}\pi x^{4}\sqrt{2}x^{3}x^{2}+x3,$$ the largest power of $x$ and hence the degree of the polynomial is 5.
name  form  degree 
Constant polynomial  $a$  0 
Linear polynomial  $ax+b$  1 
Quadratic polynomial  $ax^2+bx+c$  2 
Cubic polynomial  $ax^3+bx^2+cx+d$  3 
Basic Operations on Polynomial
 When two or more terms differ only in their numerical coefficients, we say they are similar or like terms. For example, $4x^{2}$, $\frac{3}{2}x^{2}$, and $\sqrt{3}x^{2}$ are like terms but $3x$ and $3x^{2}$ are unlike terms.

Adding and subtracting polynomials: To add or subtract polynomials, we add or subtract the coefficients of like terms. For example:
\begin{align*}
(7x^{3}6x^{2}+4x+9)+(x^{3}+4x^{2}2) & =(7+1)x^{3}+(6+4)x^{2}+(4+0)x+(92)\\
& =8x^{3}2x^{2}+4x+7
\end{align*}and
\begin{align*}
(3x^{4}8x)(5x^{3}3x+\sqrt{2}) & =(30)x^{4}+(05)x^{3}+(8(3))x+(0\sqrt{2})\\
& =3x^{4}5x^{3}5x\sqrt{2}
\end{align*} 
Multiplication of polynomials: We multiply polynomials like any other sums and simplify the result by using the exponent rule $$ax^{n}\cdot bx^{m}=ab\ x^{m+n}$$ and collect the like terms. For example,
\begin{align*}
(4x^{2}3x+5)(2x^{3}x) & =4x^{2}(2x^{3}x)3x(2x^{3}x)+5(2x^{3}x)\\
& =8x^{5}4x^{3}6x^{4}+3x^{2}+10x^{3}5x\\
& =8x^{5}6x^{4}+(104)x^{3}+3x^{2}5x\\
& =8x^{5}6x^{4}+6x^{3}+3x^{2}5x
\end{align*}It is clear that if $A$ is a polynomial of degree $n$ and $B$ is a polynomial of degree $m$, then the product $A\cdot B$ is a polynomial of degree $n+m$. In the above example, the first one is a polynomial of degree 2 and the second one is a polynomial of degree 3, and the product is a polynomial of $5=2+3$.

Division of polynomials: To divide one polynomial by another, we use long division; the procedure for such a division has been explained in detail in the next Section