1. Warm-Up
f(x)
Copy the coordinate plane with the following information.
X - X
Simplify each expression.
f(x) -
4) (x + 5) + (2x + 3)
5) (x + 9) – (4x + 6)
6) (-x2 – 2) – (x2 – 2)

2. An Intro to Polynomials
Essential Questions:
How can we identify, evaluate, add, and subtract polynomials?
How can we classify polynomials, and describe their end behavior given the function?

3. Classification of a Polynomial by Degree
Name
Example
n = 0
constant 3
n = 1
linear
5x + 4
n = 2
quadratic
2x2 + 3x - 2
n = 3
cubic
5x3 + 3x2 – x + 9
n = 4
quartic
3x4 – 2x3 + 8x2 – 6x + 5
n = 5
quintic
-2x5 + 3x4 – x3 + 3x2 – 2x + 6

4. Classification of a Polynomial by Number of Terms
Name
Example
monomial 3
binomial
5x + 4
trinomial
2x2 + 3x - 2
polynomial
5x3 + 3x2 – x + 9
binomial
-2x5 + 3x2 – x5 - 3x2 + 6
Combine like terms

5. Example 1a Classify each polynomial by degree and by number of terms.
a) 5x + 2x3 – 2x2
cubic trinomial
b) x5 – 4x3 – x5 + 3x2 + 4x3
quadratic monomial

6. Description of a Polynomial’s Graph
Example
f(x)= 3
f(x)= 5x + 4
f(x)= 2x2 + 3x - 2
f(x)= 5x3 + 3x2 – x + 9
f(x)= 3x4 – 2x3 + 8x2 – 6x + 5
f(x)= -2x5 + 3x4 – x3 + 3x2 – 2x + 6

7. Example 1b Determine the polynomial’s shape and end behavior.
a) f(x)= 5x + 2x3 – 2x2
b) f(x)= x5 – 4x3 – x5 + 3x2 + 4x3

8. Example 2
Add
(5x2 + 3x + 4) + (3x2 + 5)
= 8x2
+ 3x
+ 9

9. Example 3 -3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1 5x5 - 3x3y3 - 5xy5 5x5
Add
(-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5)
-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1
5x x3y xy5
5x5
– 3x4y3
+ 3x3y3
– 6x2
+ 1

10. Example 4 Subtract. (2a4b + 5a3b2 – 4a2b3) – (4a4b + 2a3b2 – 4ab)

11. Example 5
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) + 15
C(10) = 3000 –
C(10) = 2865
$2865

12. 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.

13. Graphs of Polynomial Functions
Graph each function below.
Function
Degree
# of U-turns in the graph
y = x3 3
y = x3 – 3x2 + 3x - 1 3
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.

14. Example 6 Graph each function. Describe its end behavior.
a) P(x) = 2x3 - 1
b) Q(x) = -3x4 + 2

15. Homework