## Polynomials~are expressions of finite length

consisting of variables or

constants either added, subtracted,

or multiplied together with

non-negative, whole number exponents.

Before we get to the standard form of a polynomial we must make sure we understand all the vocabulary that goes

along with polynomials. We will start with the word polynomial itself. The prefix "poly" means many, so that

polynomial means a collection of many things added or subtracted from one another. Now technically, this

should only apply to an expression with more than one term, but we use this word for any amount of terms in

a expression.

~Terms: are the pieces that the polynomial is actually made of. Terms may consist of constants, variables, or

a combination of the two. The only restrictions on what a term can be are:

1. The Power of each Variable must be a non-negative, whole number.

2. Each constant used must be either a integer or a rational number.

~Examples of expressions that are not polynomials (non-term expressions).

2^(3/5) x^(-1) 2.467 2x^(3/2) 2.45y

Each of these brake either restriction one or two. 2^(2/5) is not a rational number or a integer. x^(-1)

does not have a non-negative power. 2x^(3/2) does not have a whole number power. 2.45y is not a rational

number or a integer.

Now some polynomials have the same number of terms, while others have different amounts of terms in them. We

have groups for the polynomials with the same number of terms in them. Below are the common names

for these different groups.

One Term Polynomial ------------ Monomial

Two Term Polynomial ------------ Binomial

Three Term Polynomial ---------- Trinomial

Four Term Polynomial ----------- Four term Polynomial

X Term Polynomial ----------- X Term Polynomial (where X can be any natural number, except 0)

If there is a common term for a four term polynomial, then I have not heard of it. Usually higher termed

polynomials are just referred to as "Blank" Term Polynomial, where blank is the amount of terms in the

polynomial.

~Coefficients: are the constants multiplied by a variable.

~Examples of Coefficients:

the two is the coefficient --> 2x two thirds is the coefficient --> (2/3)y^2

~Example of a Non Coefficient and a Coefficient:

the 5 is not a coefficient --> 5+2x <--the two is a coefficient

~Variables: are the unknowns in the expression. They can equal any number, which means they vary from time to

time (hints the name: variable).

~Examples of variables:

X is the variable --> 2X Y is the variable --> (2/3)Y^2 Z is the variable --> Z

~Degree: of a polynomial is the highest exponent of all the terms in the expression. If a term has more than one

variable you must add the powers of the two variables in order to get the degree of that term.

~Examples of Degree:

x^2 + x +1 x^17 + x^3 + x x^(3)y^(5) + x^(6) + y^(7)

Degree is 2 Degree is 17 Degree is 8 (because 3+5=8)

~Names of degrees:

Sometimes polynomials are named by their degrees along with the number of terms. So that a second

degree trinomial would be called a quadratic trinomial. Below is a table of names with degrees

3 <-- zero degree

x + 1 <-- first degree

x^(2) + 2x + 1 <-- second degree

x^(3) + 1 <-- third degree

x^(2)y^(2) <-- fourth degree

x^(5) + x^(3) + x <-- fifth degree

x + 1 <-- first degree

x^(2) + 2x + 1 <-- second degree

x^(3) + 1 <-- third degree

x^(2)y^(2) <-- fourth degree

x^(5) + x^(3) + x <-- fifth degree

Constant

Linear Binomial

Quadratic Trinomial

Cubic Binomial

Quartic Monomial

Quintic Trinomial

Linear Binomial

Quadratic Trinomial

Cubic Binomial

Quartic Monomial

Quintic Trinomial

~Like Terms: are terms that are common in their mathematical background, or in other words are alike. These

must always be simplified in a polynomial (simplified by either adding or subtracting them).

~Examples of Like Terms:

2x+x <-- these two terms are alike so that they can be added --> 3x

1+5 <-- these two terms are alike and must be simplified to --> 6

We now know enough to write a polynomial in standard form.

The standard form of a polynomial is:

1. All term with coefficients and variables must have the coefficient first

2. You must arrange all terms according to their degree from highest to lowest

3. The leading coefficient must be positive!

3. All like terms must be simplified

~Examples of Standard Form of a polynomial:

2x^(3) + x^(2) + (5/3)x + 1 x^(4)y^(3) + y^(2) + x^(1) + 5

(leading coefficient 2) (leading coefficient 1)

Usually the coefficient in the leading term is called the leading coefficient.

The next page will discuss how to add, subtract, and multiply these standard form polynomials.