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
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
Example 1
Classify each polynomial by degree and by number of
terms.
a) 5x + 2x3 – 2x2
cubic trinomial
b) x5 – 4x3 – x5 + 3x2 + 4x3
quadratic monomial
Example 2
Add
(-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5)
-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1
5x5
- 3x3y3
- 5xy5
5x5 – 3x4y3 + 3x3y3 – 6x2
+1
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.
Example 6
Graph each function. Describe its general
shape.
a) P(x) = 2x3 - 1
a curve that always rises to the right
b) Q(x) = -3x4 + 2
a U-shape that has one turn
Polynomial Functions and Their Graphs
Objectives:
•Identify and describe the important features of the
graph of a polynomial function
•Use a polynomial function to model real-world data
Graphs of Polynomial Functions
f(a) is a local maximum if there is an interval around
a such that f(a) > f(x) for all values of x in the
interval, where x = a.
f(a) is a local minimum if there is an interval around
a such that f(a) < f(x) for all values of x in the
interval, where x = a.
Increasing and Decreasing Functions
Let x1 and x2 be numbers in the domain of a function, f.
The function f is increasing over an open interval if for
every x1 < x2 in the interval, f(x1) < f(x2).
The function f is decreasing over an open interval if for
every x1 < x2 in the interval, f(x1) > f(x2).
Example 1
Graph P(x) = -2x3 – x2 + 5x + 6.
a) Approximate any local maxima or minima to the
nearest tenth.
minimum: (-1.1,2.0)
maximum: (0.8,8.3)
b) Find the intervals over which the function is increasing
and decreasing.
increasing: x > -1.1 and x < 0.8
decreasing: x < -1.1 and x > 0.8
Exploring End Behavior of f(x) = axn
Graph each function separately. For each function,
answer parts a-c.
1) y = x2
2) y = x4
3) y = 2x2
4) y = 2x4
5) y = x3
6) y = x5
7) y = 2x3
8) y = 2x5
9) y = -x2
10) y = -x4
11) y = -2x2
12) y = -2x4
13) y = -x3
14) y = -x5
15) y = -2x3
16) y = -2x5
a. Is the degree of the function even or odd?
b. Is the leading coefficient positive or negative?
c. Does the graph rise or fall on the left? on the right?
Exploring End Behavior of f(x) = axn
Rise or Fall
a > 0
left
a < 0
right
left
right
n is even
rise
rise
fall
fall
n is odd
fall
rise
rise
fall
Example 2
Describe the end behavior of each function.
a) V(x) = x3 – 2x2 – 5x + 3
falls on the left and rises on the right
b) R(x) = 1 + x – x2 – x3 + 2x4
rises on the left and the right