# Polynomials

A** polynomial** is an algebraic expression in the form:

**P(x) = a _{n }x^{n } + a_{n - 1 }x^{n - 1 }+ a_{n - 2 }x^{n - 2} + ... + a_{1 }x^{1} + a_{0}**

** a_{n}, a_{n -1 }... a_{1} , a_{o}**... are the numbers and are called

**coefficients**.

**n** is a natural number.

**x** is the **variable**.

**a _{o}** is the

**independent term**.

### Degree of a Polynomial

The degree of a polynomial P(x) is the greatest degree of the monomials.

### Classification of a Polynomial According to Their Degree

#### Quadratic

P(x) = 2x^{2} + 3x + 2

#### Cubic

P(x) = x^{3 }− 2x^{2} + 3x + 2

#### Quartic

P(x) = x^{4} + 2x^{3}− 2x^{2} + 3x + 2

#### Quintic

P(x) = 2x^{5} − x^{4} + 2x^{3}− 2x^{2} + 3x + 2

#### Sextic

P(x) = 3x^{6} + 2x^{5} − x^{4} + 2x^{3}− 2x^{2} + 3x + 2

## Types of Polynomials

### Zero Polynomial

A polynomial that has zero as all its coefficients.

### Homogeneous Polynomial

A polynomial where all its terms or monomials are of the same degree.

P(x) = 2x^{2} + 3xy

### Complete Polynomial

A polynomial which has all the terms ordered from the greatest degree up to the independent degree.

P(x) = 2x^{3} + 3x^{2} + 5x - 3

### Ordered Polynomial

A polynomial which has its monomials ordered starting from the greatest or smallest degree.

P(x) = 2x^{3} + 5x - 3

### Equal Polynomials

Two polynomials are equal if:

1 The two polynomials have the same degree.

2 The coefficients of the terms with the same degree are equal.

P(x) = 2x^{3} + 5x − 3

Q(x) = 5x − 3 + 2x^{3}

### Similar Polynomials

Two polynomials are similar if they have the same literal part.

P(x) = 2x^{3} + 5x − 3

Q(x) = 5x^{3} − 2x − 7

### Evaluating Polynomials

Evaluating a polynomial is to find its **numerical value** when the variable x is replaced by any number.

P(x) = 2x^{3} + 5x − 3 ; x = 1

P(1) = 2 · 1^{3} + 5 · 1 − 3 = 2 + 5 - 3 = 4