Pre Calculus
Lesson #38
2016-2017
Mrs. von Stein
Name:_______________________________
Date: ________________________________
Objective  Find and use zeros of polynomial functions as sketching aids

WARM-UP
Describe the left-hand and right-hand behavior of the graph of each function.
a. f (x) =
1 2
x - 2x
4
b. f (x) = -3.6x 5 + 5x 2 -1
Zeros of Polynomial Functions
It can be shown that for a polynomial function f of degree n, the following statements
are true:
1. The function f has, at most, n real zeros.
2. The graph of f has, at most, n – 1 turning points. (Turning points, also called
relative minima or relative maxima, are points at which the graph changes from
increasing to decreasing or vice versa.)
1
Let f be a polynomial function and let a be a real number. List four equivalent statement about the
real zeros of f.
1. ___________________________________________________________________________
2. ___________________________________________________________________________
3. ___________________________________________________________________________
4. ___________________________________________________________________________
Example 1 Find the Zeros of a Polynomial Function
Find all real zeros of f(x) = -2x4 + 2x2
Then determine the number of turning points of the graph of the function.
Algebraic Solution
Graphical Solution
Use a graphing utility to graph y= = -2x4 + 2x2
The graph appears to have zeros at: ______________
The graph has ______ turning points. This is
consistent with the fact that a fourth degree polynomial
can have at most 3 turning points.
Note: because k is even, the factor -2x2 yields the repeated zero x = 0. The graph touches the x-axis
at x = 0.
2
To graph polynomial functions, you can use the fact that a polynomial function can change signs only
at it zeros. Between two consecutive zeros, a polynomial must be entirely positive or entirely
negative. This means that when the real zeros of a polynomial function are put in order, they divide
the real number line into intervals in which the function has no sign changes These resulting intervals
are test intervals.
Example 2 Sketch the graph of f(x) = 3x4 – 4x3
1. Apply the Leading Coefficient Test.
____________________________________________________________
____________________________________________________________
2. Find the Zeros of the Polynomial.
______________________________________________________________
3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test
intervals. In each test interval, choose a representative x-value and evaluate the
polynomial function.
Test interval
Representative
x-value
Value of f
Sign
Point on graph
4. Draw the Graph.
3
9
2
Example 3 Sketch the graph of f (x) = -2x 3 + 6x 2 - x
1. Apply the Leading Coefficient Test.
____________________________________________________________
____________________________________________________________
2. Find the Zeros of the Polynomial.
______________________________________________________________
3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test
intervals. In each test interval, choose a representative x-value and evaluate the
polynomial function.
Test interval
Representative
x-value
Value of f
Sign
Point on graph
4. Draw the Graph.
4
Example 4 Writing an Equation from the Graph
 Find the lowest possible degree of the polynomial
 Determine the sign of the leading coefficient
 Find all intercepts
 Determine all linear factors and their multiplicities
 Find the equation of the polynomial
5
Practice/ HW # 38
In Exercises 1 – 5 (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of
each zero and the number of turning points of the graph of the function, and (c) use a graphing utility
to graph the function and verify your answers.
1. f(x) = 3x3 -12x2 + 3x
2. f(t) = t3 -4t2 + 4t
3. g(t) = t5 -6t3 + 9t
4. f(x) = 5x4 + 15x2 + 10
5. g(x) = x3 + 3x2 – 4x – 12
In Exercises 6 - 7, find a polynomial function that has the given zeros. (There are many correct
answers).
6) 0, -2, -3
7) 4, -3, 3, 0
In Exercises 8 – 10, find a polynomial of degree n that has the given zero(s). (There are many correct
answers.
Zero(s)
Degree
8)
x = -2
n=2
9)
x = 0, 3,- 3
n=3
10)
x = -5, 1, 2
n=4
6
In Exercises 11 – 14, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b)
finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous
curve through the points.
11) f (x) =
1 2
(t - 2t +15)
4
12) f (x) = x 3 - 3x 2
7
13) f (x) = 3x 3 -15x 2 +18x
14) f (x) = -5x 2 - x 3
15) Find the equation of the polynomial




Find the lowest possible degree of the polynomial
Determine the sign of the leading coefficient
Find all intercepts
Determine all linear factors and their multiplicities
8
Solutions to Odd #s
1.
3.
5.
7.
9.
11.
13.
f(x) = ½(x+4)(x + 2)2(x-1)
9