Polynomials: Basics

A. Definition of a Polynomial

A polynomial is a combination of terms containing numbers and variables raised to positive
( or zero ) whole number powers.

Examples of Polynomials

NOT polynomials

	( power is a fraction )
	(power is a fraction)

B. Termino logy

1. Degree

a. Term Degree : sum of powers in a term

	the degree is 8
	the degree is 4
4	the degree is 0

b. Polynomial Degree : maximum (not total) term degree

	the degree is 8
	the degree is 5

2. Descending Order

We often write polynomials in order from the highest term degree to the the lowest.

For instance, we rewrite as
 

C. Adding / Subtracting Polynomials

We combine like terms as before.

Beware: minus signs and parentheses

1. Find

2. Find

D. Multiplying Polynomials By Monomials

A monomial is a one -term polynomial. Use the distributive property .

Find

E. Multiplying Binomials

A binomial is a two-term polynomial.

Method 1: Distributive Property

If the problem is to expand , we distribute the to the
two terms of the second binomial:

Now use the distributive property again to get


A shortcut to the above method is called FOIL

Method 2: FOIL

FOIL is an acronym for “First-Outer-Inner-Last”

Consider the fol lowing example :

Find: using FOIL

First:

Outer:

Inner:
 
Last:

Thus we get

F. Multiplying Polynomials of Any Size

Method 1: Distributive Property

If the problem is to expand , we distribute the
to the terms of the second polynomial

Now use the distributive property again

Thus, after combining like terms, we get

A shortcut to the above method is called the factor table

Method 2: Factor Table

You make a “tic-tac-toe” grid, and fill in the boxes with the products.

Consider

Make factor table:

Then fill in the table with the products:

Collecting like terms: