Bell-ringer

Holt Algebra II text page 431 #72-75, 77-80

7.1 Introduction to Polynomials

Definitions

Monomial - is an expression that is a number, a variable, or a product of a number and variables. i.e. 2, y, 3x, 45x2…

Constant - is a monomial containing no variables. i.e. 3, ½, 9 …

Coefficient - is a numerical factor of a monomial. i.e. 3x, 12y, 2/3x3, 7x4 …

Degree - is the sum of the exponents of a monomial’s variables. i.e. x3y2z is of degree 6 because x3y2z1 = 3 + 2+ 1 = 6

Definitions

Polynomial- is a monomial or a sum of terms that are monomials. These monomials have variables which are raised to

whole-number exponents.

The degree of a polynomial is the same as that of its term with the greatest degree.

Examples v. Non-examples

Examples

5x + 4

x4 + 3x3 – 2x2 + 5x -1

√7x2 – 3x + 5

Non – examples

x3/2 + 2x – 1 3/x2 – 4x3 + 3x – 13 3√x +x4 +3x3 +9x

+7

Classification

We classify polynomials by…

…the number of terms or monomials it contains

AND

… by its degree.

Classification of Polynomials

Classifying polynomials by the number of terms…

monomial: one term binomial: two terms trinomial: three terms Poylnomial: anything with four or more

terms

Classification of a Polynomial

Degree Name Example

-2x5 + 3x4 – x3 + 3x2 – 2x + 6

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

constant 3

linear 5x + 4

quadratic 2x2 + 3x - 2

cubic 5x3 + 3x2 – x + 9

quartic 3x4 – 2x3 + 8x2 – 6x + 5

quintic

Compare the Two Expressions

How do these expressions compare to one another?

3(x2 -1) - x2 + 5x and 5x – 3 + 2x2

How would it be easier to compare?

Standard form - put the terms in descending order by degree.

Examples

Write each polynomial in standard form, classifying by degree and number of terms.

1). 3x2 – 4 + 8x4

= 8x4+ 3x2 – 4 quartic trinomial2). 3x2 +2x6 - + x3 - 4x4 – 1 –x3 = 2x6- 4x4 + 3x2 – 16th degree polynomial with four terms.

Adding & Subtracting Polynomials

To add/subtract polynomials, combine like terms, and then write in standard form.

Recall: In order to have like terms, the variable and exponent must be the same for each term you are trying to add or subtract.

Examples

Add the polynomial and write answer in standard form.

1). (3x2 + 7 + x) - (14x3 + 2 + x2 - x) = =- 14x3 + (3x2 - x2) +(x -x) + (7- 2) = - 14x3 + 2x2 + 5

Example

Add

(-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5)

-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 15x5 - 3x3y3 - 5xy5

5x5 – 3x4y3 + 3x3y3 – 6x2 + 1

Example

Subtract.

(2x2y2 + 3xy3 – 4y4) - (x2y2 – 5xy3 + 3y – 2y4)

= 2x2y2 + 3xy3 – 4y4 - x2y2 + 5xy3 – 3y + 2y4

= x2y2 + 8xy3 – 2y4 – 3y

Evaluating Polynomials

Evaluating polynomials is just like evaluating any function.

*Substitute the given value for each variable and then do the arithmetic.

Application

The cost of manufacturing a certain product can be approximated by f(x) = 3x3 – 18x + 45, where x is the number of units of the product in hundreds. Evaluate f(0) and f(200) and describe what they represent.

f(0) = 45 represents the initial cost before manufacturing any products f(200) = 23,996,445 represents the cost of manufacturing 20,000 units of the product.

Exploring Graphs of Polynomial Functions Activity

Copy the table on page 427Answer/complete each question/step.

Graphs of Polynomial Functions

Graph each function below.

Function Degree # of U-turns in the graph

y = x2 + x - 2 2 1

y = 3x3 – 12x + 4 3 2

y = -2x3 + 4x2 + x - 2 3 2

y = x4 + 5x3 + 5x2 – x - 6 4 3

y = x4 + 2x3 – 5x2 – 6x 4 3

Make a conjecture about the degree of a function and the # of “U-turns” in the graph.

Graphs of Polynomial Functions

Graph each function below.

Function Degree # of U-turns in the graph

y = x3 3 0

y = x3 – 3x2 + 3x - 1 3 0

y = x4 4 1

Now make another conjecture about the degree of a function and the # of “U-turns” in the graph.

The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.

Now You

Graph each function. Describe its general shape.

P(x) = -3x3 – 2x2 +2x – 1 An S-shaped graph that always rises on

the left and falls on the right.

Q(x) = 2x4 – 3x2 – x + 2 W-shape that always rises on the right and

the left.

Check Your Understanding

Create a polynomial.

Trade polynomials with the second person to your left.

Put your new polynomial in standard form then……identify by degree and number of terms …identify the number of U - turns.

Turn the papers in with both names.

Homework

Page 429-430 #12-48 by 3’s.