7.1 An Intro to Polynomials7.1 An Intro to Polynomials7.1 An Intro to Polynomials7.1 An Intro to PolynomialsObjectives: •Identify, evaluate, add, and subtract polynomials•Classify polynomials, and describe the shapes of their graphs

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

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Example 1Classify each polynomial by degree and by number of terms. Remember to simplify first.

a) 5x + 2x3 – 2x2 cubic

b) x5 – 4x3 – x5 + 3x2 + 4x3

quadratic

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

trinomial

monomial

Example 2Polynomials are used in business to express the cost of manufacturing products. If the cubic function

C(x) = 3x3 – 15x + 15 gives the cost of manufacturing x units (in thousands) of a product, what is the cost to manufacture 10,000 units of the product?

C(x) = 3x3 – 15x + 15

C(10) = 3(10)3 – 15(10) + 15C(10) = 3000 – 150 + 15

C(10) = 2865

$2865Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Collins Type I

Using at least three lines, describe how you identify like terms and why they are important.

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Example 3Add

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

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

5x5 – 3x4y3

+ 3x3y3

– 6x2 + 1

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Example 4Subtract.

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

= 2x2y2 + 3xy3 – 4y4

- x2y2 + 5xy3 – 3y + 2y4

= x2y2+ 8xy3

– 2y4 – 3y

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Graphs of Polynomial Functions

• You will need:– A Partner– A Graphic Calculator– A Textbook

• Page 427 Activity– Answer all questions in your notebook– Each person should record the findings

• You have 7 minutes

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

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.

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

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.

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Example 5Graph each function. Describe its general shape.

a) P(x) = 2x3 - 1

b) Q(x) = -3x4 + 2

a curve that always rises to the right

a U-shape that falls to both sides

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Test Your Might

Sketch the graph of a cubic function that intersects the x-axis at exactly the number of points indicated on your card. Write impossible if appropriate.

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs

Homework

Lesson 7.1 exercises 17-27 odd,

33, 35, 43, 45, 51-59 odd, 81

Objectives: Identify, evaluate, add, and subtract polynomialsClassify polynomials, and describe the shapes of their graphs