Polynomials

A polynomial is an algebraic expression in the form:

P(x) = an xn + an - 1 xn - 1 + an - 2 xn - 2 + ... + a1 x1 + a0

an, an -1 ... a1 , ao... are the numbers and are called coefficients.

n is a natural number.

x is the variable.

ao 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) = 2x2 + 3x + 2

Cubic

P(x) = x3 − 2x2 + 3x + 2

Quartic

P(x) = x4 + 2x3− 2x2 + 3x + 2

Quintic

P(x) = 2x5 − x4 + 2x3− 2x2 + 3x + 2

Sextic

P(x) = 3x6 + 2x5 − x4 + 2x3− 2x2 + 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) = 2x2 + 3xy

Complete Polynomial

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

P(x) = 2x3 + 3x2 + 5x - 3

Ordered Polynomial

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

P(x) = 2x3 + 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) = 2x3 + 5x − 3

Q(x) = 5x − 3 + 2x3

Similar Polynomials

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

P(x) = 2x3 + 5x − 3

Q(x) = 5x3 − 2x − 7

Evaluating Polynomials

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

P(x) = 2x3 + 5x 3 ; x = 1

P(1) = 2 · 13 + 5 · 1 − 3 = 2 + 5 - 3 = 4