1. Introduction to Polynomials
24 February 2011

2. What is a polynomial?
polynomial

3. Definition of Polynomial
A polynomial is algebraic expression that
can be written in the form
anxn + an-1xn-1 + … + a2x2 + a1x1 + a0
An equation or an expression with a single variable raised to (usually many) powers
All exponents are whole numbers
an ≠ 0

4. Key Features of Polynomials
Degree (n, greatest exponent in polynomial)
anxn + an-1xn-1 + … + a2x2 + a1x1 + a0
Leading Coefficient (an)
Constant (the term without a variable)

5. Notes About Polynomials
Exponents can be skipped!
Ex. 8x8 – x4 + 2x – 1
Written in numerical order from greatest exponent to least exponent
Ex. 2x3 + 7x8 – 14x2 – x5
Rewrite: 7x8 + 5x5 + 2x3 – 14x2 – 12

6. Classifying Polynomials
Classified according to key features
Degree
Leading Coefficient
Constant
Degree especially important!

7. Naming Polynomials Named according to their degree
Ex. 3x11 + 7x5 – 11x3 + 2x2 + 14x – 63
11th degree polynomial
Some polynomials have special names:
0 degree (just a constant term): Constant
1st degree: Linear
2nd degree: Quadratic
3rd degree: Cubic
4th degree: Quartic
5th degree: Quintic

8. Examples: x3
Degree:
Name:
4x x7 + 5x2 – 7x
Rewrite:

9. Examples:
0x9 + 2x6 + 3x7 + x8 – 2x – 4
Rewrite: x8 + 3x7 + 2x6 – 2x – 4
Degree: 8
Name: 12
Degree:

10. Your Turn:
On the “Introduction to Polynomials” Handout, find the table labeled “Classifying Polynomials”.
Identify the degree, leading coefficient, and constant of each of the polynomials in the table.

11. Your Turn:
Go back to the “Classifying Polynomials” table. Name each polynomial.

12. Multiplication of Polynomials
Monomial by anything = Distribute
Binomial by binomial = FOIL
Multiplying with trinomials (or larger) = Variation of FOIL
Make sure you distribute each term in the 1st set of parentheses to every term in the 2nd set of parentheses

13. Example #1
(– 4x2 – 5x – 1)(4x2 – 6x – 2)

14. Example #2
(x – 2)3

15. Your Turn:
On the “Introduction to Polynomials” handout, complete problems 1 – 14. Remember to simplify and to rewrite the polynomials from greatest exponent to least exponent!