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
