Polynomials: An algebraic expression is an expression which is made up of variables and constants along with some algebraic operations. The several parts of an algebraic expression seperated by + or – operations are called the terms of the expression. e.g.

(i) $ax^{3}+by^{2}x+cz^{2}$

is an algebraic expression with three terms and three variables $x,y,z$ .

(ii) $x^{2}+2xy+y^{2}$ is an algebraic expression with three terms and two variables $x,y$ .

(iii) $2\sqrt{x}+3x$ is an algebraic expression with two terms and one variable $x$ .

(iv) $\frac{1}{x}$ is an algebraic expression with one terms and one variable.

In an algebraic expression , if the powers of variables are non-negative integers , then it is a polynomial.

In the above examples , (i) and (ii) are polynomials, where as (iii) and (iv) are not polynomials.

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Polynomials in one variable :

Polynomials in one variable are algebraic expressions that consists of terms in the form of $ax^{n}$ , where $n$ is non-negative integer and a is constant . e.g. $x^{3}, \, 3x^{2}+2x-2, \, -7x^{6}+10 ,$ all are polynomials in variable $x$ .

Each term of a polynomial has a coefficient . so in $-x^{3}+\sqrt{2}x+3$ , the coefficient of $x^{3}$ is -1, coefficient of $x$ is $\sqrt{2}$ and coefficient of $x^{0}$ is 3.

Degree of a polynomial:

The degree of a polynomial in a single variable $x$ is the highest power of $x$ in its expression. e.g. $7x^{5}-4x^{2}+5$ is a polyn0mial of degree 5 and $x^{6}-100$ is a polynomial of degree 6.

In general any polynomial of degree $n$ is an expression of the form

$\large p(x)= a_{0}+a_{1}x+a_{2}x^{2}+............+a_{n-1}x^{n-1}+a_{n}x^{n},$ where $\large a_{0},\, a_{1}, \, a_{2},.........., a_{n-1} , a_{n}$ are constants , $a_{n}\neq 0$ and $n$ is a non-negative integer .

Here $a_{0}$ is called the constant term of the polynomial and $a_{1}, a_{2},a_{},............,a_{n-1},a_{n}$ are called the coefficient of $x,x^{2}, x^{3},..........., x^{n-1} \, \, and \, \, x^{n}$ respectively.

In particular if all the constants are zero , then we get $\large p(x)=0$ , the zero polynomial. Zero polynomial has no non-zero terms so the degree of zero polynomial is not defined.

Constant polynomial:

A polynomial containing only the constant term is called constant polynomial. e.g. $\sqrt{2}, 3,-7$ all are constant polynomials.

Let $p(x)=a\, ,\, a\neq 0$ is a non-zero constant polynomial ,

then $p(x)$ can be written as $p(x)=ax^{0}$

$\Rightarrow deg\, \, p(x)=0$

Therefore the degree of any non-zero constant polynomial is zero.

Based on the number of terms, polynomials are classified as

(i) A polynomial containing one term is called a monomial. e.g. $5x^{3}, \, -9x, \, 7$ etc. all are monomials.

(ii) A polynomial containing two terms is called a binomial. e.g. $9x^{2}+2,\, \, -7x^{5}+3x$ etc.

(iii)A polynomial containing three terms is called a trinomial. e.g. $\sqrt{5}x^{3}+7x-2, \, \, 2x^{3}+7x+9, \, x^{25}-10x+5$ etc. all are trinomials.

Linear polynomial :

A polynomial of degree one is called a linear polynomial. A linear polynomial in $x$ is of the form

$\large p(x)=ax+b, \: \, where \, \, a, \, b \, \, are \, \, constants \, \, and \,\, a \neq 0$ .

Any linear polynomials in $x$ have at most two terms . e.g. $7x+2, \, -2x,\, \sqrt{2}x-6$ all are linear polynomials.

Quadratic polynomial :

A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial in one variable will have at most tree terms. Any quadratic polynomial in $x$ will be of the form $\large ax^{2}+bx+c\, \, where \, \, a\neq 0\, \, and \, \, a,b,c\, \, are\, \, constants.$

e.g. $4x+ 5x^{2}, x^{2}, x^{2}-x+6$ all are quadratic polynomials.

Cubic polynomial :

A polynomial of degree 3 is called cubic polynomials. Any cubic polynomial can have at most 4 terms. $6x^{3}-6x, 2x^{3}+3x^{2}-5x+2, 4x^{3} , x^{3}-2x^{2}$ all are examples of cubic polynomials.