4 minutes

Warm-Up

Evaluate each expression for x = -2.

1) -x + 1 2) x 2 - 5 3) -(x – 6) Simplify each expression.

4) (x + 5) + (2x + 3) 5) (x + 9) – (4x + 6) 6) (-x 2 – 2) – (x 2 – 2)

7.1 An Intro to Polynomials

Objectives:

•Identify, evaluate, add, and subtract polynomials •Classify polynomials, and describe the shapes of their graphs

Classification of a Polynomial

Degree n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 Name constant linear quadratic cubic quartic quintic Example 3 5x + 4 2x 2 + 3x - 2 5x 3 + 3x 2 – x + 9 3x 4 -2x 5 – 2x + 3x 4 3 + 8x – x 3 2 – 6x + 5 + 3x 2 – 2x + 6

Example 1

Classify each polynomial by degree and by number of terms.

a) 5x + 2x 3 – 2x 2 b) x 5 – 4x 3 – x 5 + 3x 2 + 4x 3 cubic trinomial quadratic monomial

Example 2

Add (5x 2 + 3x + 4) + (3x 2 + 5) = 8x 2 + 3x + 9

Example 3

Add (-3x 4 y 3 + 6x 3 y 3 – 6x 2 + 5xy 5 + 1) + (5x 5 – 3x 3 y 3 – 5xy 5 ) 5x 5 5x 5 -3x 4 y 3 + 6x 3 y 3 - 3x 3 y 3 – 6x 2 + 5xy 5 - 5xy 5 + 1 – 3x 4 y 3 + 3x 3 y 3 – 6x 2 + 1

Example 4

Subtract.

(2a 4 b + 5a 3 b 2 – 4a 2 b 3 ) – (4a 4 b + 2a 3 b 2 – 4ab) 2a 4 b + 5a 3 b 2 -4a 4 b - 2a 3 b 2 – 4a 2 b 3 + 4ab -2a 4 b + 3a 3 b 2 – 4a 2 b 3 + 4ab

Example 5

C(x) = 3x 10,000 units of the product?

3 – 15x + 15 the cost of manufacturing x units (in thousands) of a product, what is the cost to manufacture C(10) = 3(10) 3 – 15(10) + 15 C(10) = 3000 – 150 + 15 C(10) = 2865 $2865

Graphs of Polynomial Functions

Graph each function below.

Function y = x 2 + x - 2 y = 3x 3 y = -2x 3 – 12x + 4 + 4x 2 + x - 2 y = x 4 + 5x 3 + 5x 2 – x - 6 y = x 4 + 2x 3 – 5x 2 – 6x Degree # of U-turns in the graph 2 3 3 4 4 3 3 1 2 2 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 y = x 3 y = x 3 – 3x 2 y = x 4 + 3x - 1 Degree # of U-turns in the graph 3 3 4 0 0 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.

Example 6

Graph each function. Describe its general shape.

a) P(x) = 2x 3 - 1 a curve that always rises to the right b) Q(x) = -3x 4 + 2 a U-shape that has one turn

Homework

p.429 #17-21 odds,29,33,39-47 odds,51-57 odds