What are Polynomials?

A Polynomial is an algebraic expression with non-negative integral powers.
Let us understand this in more detail.

As you may be aware, an algebraic expression is made up of constants and variables.

• A symbol having a fixed numerical value is called a constant
  For example 2, -7, 0.05, $\frac{3}{7}$ are all constants. These are generally denoted as $a,b,c$, etc.
  
• A symbol which can take any real value is called a variable
  These are generally denoted by letters such as $x,y,z$, etc. An algebraic expression is made up by combining constants and variables
  Example $5x + \frac{5}{6}y - 9$ is an algebraic expression.
  
  

Different parts of an algebraic expression are joined by $+$ or $-$ operators.

Notice that in the definition of algebraic expression, we said that it should contain non-negative integral powers. Read this as the terms of polynomials should have variables with their powers as whole numbers.

Examples

• $5x + 2y - 120$ is a polynomial


• $5x + 2y^{-1} - 120$ is not a polynomial as the power of variable y is -1


• $y + 2z - \dfrac{1}{2}$ is a polynomial


• $y^{\frac{6}{7}} + 2z - \dfrac{1}{2}$ is not a polynomial as power of y is $\dfrac{6}{7} $ which is not a whole number or a non-negative integer.

Please note that in the expression $5x - 2y - 120$, 5 and -2 are coefficients of $x$ and $y$ respectively and -120 is a constant.

Refer Polynomials Solved Examples here

Degree of a Polynomial

In case of a polynomial in one variable, the degree of a polynomial is the highest power of the variable.
Examples

• $5x + 2$ is a polynomial in $x$ of degree 1 as the highest power of $x$ is 1.
• $y^3 + 2y^2 + 5y + 4$ is a polynomial in $y$ of degree 3 as the highest power of $y$ is 3.

In case of a polynomial in two or more variables, the degree of a polynomial is found by taking the sum of the powers of the variables in each term and then taking the highest sum so obtained.
Examples

• $5x^2y^3 + 2x^2y^2 - 8xy$ is a polynomial in $x$ and $y$ of degree 5.
• $y^3 + 2x^2y^2 + 5xy + 4$ is a polynomial in $x$ and $y$ of degree 4.

Remainder Theorem

If a polynomial $f(x)$ of degree n >= 1, is divided by $(x - a)$, then the remainder is $f(a)$.
$f(a)$ is obtained by putting $a$ as value of variable $x$ in polynomial $f(x)$.

Example
Find the remainder when $f(x)$ = $x^3 - 2x^2 + 5$ is divided by $(x - 2)$

Solution:
$(x - 2) = 0$ ⇒ $x$ = 2
By the Remainder Theorem, we know that if a polynomial $f(x)$ is divided by $(x - 2)$, the remainder is $f(2)$.
Hence finiding $f(2)$ = $2^3 - 2*2^2 + 5$ = $5$.

Factor Theorem

In Remainder Theorem, we saw that if a polynomial $f(x)$ of degree n >= 1, is divided by $(x - a)$, then the remainder is $f(a)$.
If $f(a)$ so obtained is 0, then $(x - a)$ is a factor of $f(x)$.

The reverse is also true.
If $(x - a)$ is a factor of $f(x)$, then $f(a)$ = 0

Example
Show that $(x - 2)$ is a factor of $f(x)$ = $x^3 - 2x^2$

Solution
By the Factor Theorem, $(x - 2)$ is a factor if $f(2)$ = 0
Now, $f(x)$ = $x^3 - 2x^2$
⇒ $f(2)$ = $2^3 - 2*2^2$ = $0$.
Hence, $(x - 2)$ is a factor of $f(x)$ = $x^3 - 2x^2$

Algebraic or Polynomial Identities

An equation that is true for every value of the variable is called an identity.
For factorization of polynomials, we use the following identities.