Polynomials

Algebraic Transformations

Algebraic Expressions

Basic Definitions

Polynomials

Algebraic Transformations

Outline

Factoring

Factoring Quadratic Polynomials

Factoring Cubic Polynomials

Factor Theorems

Other Transformations

Expanding

Completing Perfect Square

Rationalizing Denominators

Polynomials

Key Topics Remaining: Algebraic Transformations » Factoring » Factoring Quadratic Polynomials » Factoring Cubic Polynomials » Factor Theorens » Expanding » Completing Perfect Square » Rationalizing Denominators

A polynomial is an algebraic expression involving variables with only non-negative integer exponents.
The degree of a polynomial is determined by the term with the highest degree. For example, the expression

4 x5 y3 – 7 x3 y2 z + 2 x y + 3

is a polynomial of degree 8.

A polynomial of n-th degree with a single variable is an algebraic expression of the form

Pn(x) = an xn + an – 1 xn – 1 + an – 2 xn – 2 + ... + a1 x + a0,

where x is the variable and an ≠ 0.
The numbers ak (k = 0, 1, 2, ..., n) are called the coefficients of the polynomial; in particular, a0 is the constant coefficient.

A polynomial is said to be a monic polynomial, if the coefficient of the leading term equals unity, an = 1.

Some polynomials have special names.
A polynomial containing only a single term is called a monomial.
A polynomial with two terms is a binomial.
A polynomial with three terms is a trinomial.

One can say that a polynomial is an additive combination of monomials.

Polynomials can also be classified according to their degrees.

A linear polynomial is a polynomial of the first degree:

P1(x) = a1 x + a0.

A quadratic polynomial is a polynomial of the second degree:

P2(x) = a2 x2 + a1 x + a0.

A cubic polynomial is a polynomial of the second degree:

P3(x) = a3 x3 + a2 x2 + a1 x + a0.

Polynomials are very important, since many mathematical and physical problems are formulated in terms of polynomial equations.
One can easily manipulate or evaluate polynomials. However, it is not so easily to find the roots of a polynomial of the third or higher degree.