# Polynomials

## Definition of Polynomials

The Polynomials may be defined as the Algebraic Expressions in which the Exponent or Power of all the variables present in the expression is a Whole Number. If the Exponent or Power of any variable is a Fraction or Negative, then, the Algebraic expression can not be considered as a polynomial.

## Symbol of polynomials

If the variable of a Polynomial is $x$, then the polynomial can be written as $p(x)$, $q(x)$, or $r(x)$. Similarly, if the variable present in a Polynomial is $y$, then, the polynomial is written as $p(y)$, $q(y)$, or $r(y)$

$p(x) = {x^2} + 2x + 3$
$q(x) = {x^4} + 9$
$r(x) = {x^2} - 9x + 4$

## Example of Polynomials

• $p(x) = x + 1$

The above expression has only one variable and the power of the variable is $1$ which is a whole number. Therefore, the above expression is Polynomial.

• $p(x) = {x^4} + 4{x^3} + 3{x^2} + 2x + 1$

Similarly, the Exponents or Powers of variables in the above expression are $4\,, 3\,, 2\,,$ and $1$ for $1st\,, 2nd\,, 3rd\,,$ and $4th$ terms respectively. All the powers are Whole Numbers. Therefore, the given Algebraic Expression is a Polynomial.

• $p(x) = x + \frac{1}{x}$

The above algebraic expression can also be written as $x + {x^{ - 1}}$. Here, the Power of the $1st$ term is $1$, and the second term is $-1$. The power of the second term is Negative. Therefore, the above Algebraic Expression is not a Polynomial.

Algebraic expressions are formed by constants and variables. For example, the algebraic expression ${x^2} + 2y + 3$ has variables x and y and constants 1, 2, and 3 multiplied with first, second, and third terms respectively.

Now, let’s assume that we want to write an expression, but we do not know the constants. In that case, we denote the constants as a, b, c, etc. However, even if we denote constants with a, b, c, etc., the value of the constant remains the same throughout the time. But, the values of the variables which are denoted as x, y, z, etc. change various times while solving an equation. The value of the variables depends on each other.