Polynomials Explained With Examples

What is Polynomial?

The word polynomial is derived from the Greek words:

‘poly’ that means ‘many’

and ‘nominal’ means ‘terms’,

So, altogether it means“many terms”.

A polynomial can have any number of terms but not infinite.

Definition of Polynomials 

Polynomials are the algebraic expressions that consist of variables and coefficients. We can perform all the arithmetic operations like addition, subtraction, multiplication and also taking positive integral power for polynomial expressions except division by variable.

Composition:

1.Constants:

E.g. 1, 2, 3 so on

2.Variables:

E.g. x, y, z etc.

Exponents:

E.g. x⁵, x⁹, x³ etc.

Notation

A polynomial in the variable “x” is the algebraic expression of the form

P(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + …… + a₁x + a. (a_n 0)

Here,

• “n” is the highest power of “x”.
• “a_n” is co-efficient of variables, a real number.

An example of Polynomials

P(x) = 6x³ + 4x² + 2x + 5

n = 3, the highest power. and a_n = 6, 4, 2and 5 are the co-efficient of variables x³, x², x¹, x respectively.

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of “x” in any polynomial.

Example:

In this Polynomial,

P(x) = 6x3 + 4x2 + 2x + 5

The highest power of variable i.e. x is 3. That’s why the degree of polynomial P(x) = 6x³ + 4x² + 2x + 5

is 3.

Leading Co-efficient of polynomial

The co-efficient of highest power of variable in a polynomial is called the leading co-efficient of polynomial.

Example:

In this polynomial,

P(x) = 6x3 + 4x2 + 2x + 5

The co-efficient of highest power i.e. x3 is 6. That’s why the leading coefficient of polynomial P(x) = 6x³ + 4x² + 2x + 5is 6.