Download Example: Example:

 Example: Example: Example: Zeros of a Polynomial—AKA X-Intercepts; AKA Roots; AKA Solutions: ___________
____________________________________________________________
The ______________________ will tell you how many zeros the polynomial has. # of zeros = Multiplicity—how many of the same factors occur. Even Multiplicity: graph touches and turns at the zero. Odd Multiplicity: graph crosses the zero. Example: If g(5) = 0, what point is on the graph of g?
The point __________ is on the graph of g. (Type an ordered pair).
The corresponding x-intercept of the graph of g is ______. (Type an integer or a
fraction.)
Example: Form a polynomial whose zeros and degree are given.
Zeros: -4, multiplicity 1;
-1, multiplicity 2;
degree 3
The polynomial has integer coefficients and a leading coefficient of 1.
Example:
Graphing Polynomials—
1. Leading Coefficient Test
2. Possible Turning Points of a Polynomial Degree of the Polynomial – 1 3. Behavior Near Zero (x-­‐ intercept) a) Substitute a zero for a factor that does not make f(x) = 0. b) Simplify c) The zero will take the behavior of the simplified polynomial. 4.
5.
6.
7.
Find x-­‐ and y-­‐ intercept Determine multiplicity Determine parent function Determine behavior for large values of x Find all seven properties of f (x) = x 2 (x − 9) . How to find MAX/MIN on Calculator: Section 4.2: Properties of Rational Functions

Find the Domain of a Rational Function
The denominator of a fraction can NOT equal the value of ___________________________. Example: Find the domain of the following functions in set builder and interval notation: 2
a) g(x) =
x+3
b) h(x) =
6
x +5
2
5
c) f (x) = 2
x +16x + 39 
I.
Find Asymptotes of Rational Functions Vertical Asymptotes •
•
Graphs CAN NOT touch or cross a vertical asymptote. If you have the same factor in the numerator and denominator…it is NOT a V.A., it is a hole. How to find a V.A.? Set the denominator equal to zero. II.
Horizontal Asymptotes If n – m = 1, you have an oblique asymptote (slant asymptote). To find: Long or Synthetic Division. Remember you just want the quotient….not the remainder. Example: If a rational function is proper, then ________________ is a horizontal asymptote. 6x 2 + 5x − 6
Example: Find Vertical and Horizontal/Oblique Asymptote if any of R(x) =
. 2x + 3
Example: Find the Vertical and Horizontal/Oblique Asymptote if any of
x 3 − 64
f (x) = 2
.
x − x − 12

Transformations of Rational Functions
Example:
4.3 The Graph of a Rational Function

Analyzing the Graph of a Rational Function
1. Factor numerator and denominator. Find the domain of the Rational Function.
2. Locate intercepts of f(x).
x-intercepts: set numerator equal to zero
y-intercepts: find f(0).
3. Find Asymptotes
Vertical asymptotes: set the denominator equal to zero (watch for holes!)
Horizontal asymptotes: 3 rules
Oblique asymptotes: Long or Synthetic Division
4. Plot points to make graph OR
Determine behavior at each vertical asymptote
a) substitute VA in every x-value except where it will make the function
undefined.
b) Simplify
c) Match behavior based on 4 graphs
Example: f (x) =
x + 14
x(x + 18)
Example: g(x) =
x 2 − 15x − 16
x + 19
35x 2 − 18x − 81
Example: h(x) =
5x 2 − 54x + 81
Example: A parcel delivery service has contracted you to design a closed box with a
square base that has a volume of 10,500 in3.
a) Express the surface area of the box as a function of x and simplify.
b) Use a graphing utility to graph s(x). [0, 100, 10] [0, 8000, 1000]
c) What is the minimum amount of material that can be used to construct the box?
d) What are the dimensions of the box that minimize the surface area?
4.4 Polynomial & Rational Inequalities
WARM UP: What is the domain of g(x) =

x−6
?
x+4
Solve Polynomial and Rational Inequalities Steps
1. Get the equation in the form: f (x) < 0; f (x) > 0; f (x) ≤ 0; f (x) ≥ 0
2. Polynomial: factor in order to determine critical values
Rational: combine so there is one fraction. Remember set the numerator and
denominator equal to zero for critical values.
3. Make a # line highlighting the critical values.
4. Pick test values in each interval and test those values in the equation from step 2.
* f(x) > 0 then +++++
* f(x) < 0 then ---------5. Determine needed interval; Graph solution.
* Remember that you do not put [ ] around V.A.’s OR infinity
symbols.
Example: Solve the inequality x 2 − 4x > 12
Example: Solve the inequality x 4 > 4x 2
x 2 (9 + x)(x − 7)
Example: Solve the following inequality.
≥0
(x + 4)(x − 5)
Example: Solve the inequality:
2x
≥1
x +1
Example: Solve the inequality:
x +1
≤2
x −1
Example: Suppose the daily cost C of manufacturing “x” bicycles is given by C(x) =
40x + 4250. Now the average daily cost is given by C(x) =
40x + 4250
. How many
x
bicycles must be produced each day for the average cost to be no more then $90?
4.5 The Real Zeros of A Polynomial Equation
WARM UP: Factor –
a) 3x 2 + 2x − 1
b) 8x 2 + 6x − 5
Types of Real Zeros—
1. Rational Zeros: _______________________________________________
Examples:
2. Irrational Zeros: ______________________________________________
Examples:

Use the Remainder and Factor Theorem
Example: Determine whether (x – 3) is a factor of 3x 4 − 7x 3 − 14x − 12
Example: Determine whether (x – 1) is a factor of 4x 4 − 2x 3 + 7x − 9

Use the Rational Zeros Theorem to list the Potential Rational Zeros of
a Polynomial Function
Example: Find the Possible Rational Zeros for f (x) = 6x 4 − x 2 + 9 .
If the equation does not factor easily…
To find the Real Zeros of a Polynomial—
1.
2.
3.
4.
Determine how many zeros the polynomial could have—
Determine the PRZ’s
Graph the equation using a graphing utility
Extract all zeros that are integers/clearly visible using synthetic until a quadratic
or linear equation is left.
5. Find the last zero(s) by factoring, quadratic formula and/or solving linear
equations
−b ± b 2 − 4ac
NOTE: Quadratic Formula: x =
2a
Example: Find the real zeros of:
a) 3x 3 − x 2 + 3x − 1
b) 3x 4 − 25x 3 + 71x 2 − 75x + 18
c) 3x 4 − 58x 3 + 280x 2 − 483x + 117

Use the Theorem for Bound on Zeros
Example: Find a bound on the real zeros of the polynomial function.
a) f (x) = 3x 4 + 6x 3 − x 2 − 12x − 15
b) f (x) = 2x 4 + 2x 3 − x 2 − 6x − 2

The Intermediate Value Theorem
Example: Use the Intermediate Value Theorem to show that the polynomial function has a zero in the given interval. f (x) = 5x 3 + 5x 2 − 8x + 9; [−5,−1] 
Graphing Polynomials using a graphing utility
a) f (x) = x 3 + 2x 2 − 5x − 6
[-10, 10, 1] x [-38, 44, 5]
b) g(x) = x 4 + x 3 − 3x 2 − x + 2
[-8, 8, 1] x [-42, 42, 100]
4.6 Complex Zeros; Fundamental Theorem of Algebra
Example: Degree 3; Zeros: 2, -9 – i. What is the other zero?

Find Polynomials with specified zeros steps
1.
2.
3.
4.
Make sure that all zeros are accounted for by referencing degree.
List all of the factors.
Expand equation using FOIL/Distributive Method
Determine Leading Coeffiicient
NOTE: i = −1
i 2 = −1
i 3 = −i
i4 = 1
Example: Form a polynomial f(x) with real coefficients having the given degree and
zeros.
a) Degree 4; zeros: 2, multiplicity 2; 6i
b) Degree 4; zeros: -2, multiplicity 2; 3 – 5i
Theorem: Every polynomial function of odd degree with real coefficients will have at
least one real zero.

Find the Complex Zeros of a Polynomial Function
Example: Find the complex zeros of the following equations—
a) f (x) = x 2 − 18x + 90
b) g(x) = 3x 4 − 10x 3 − 12x 2 + 122x − 39 (Write answer in factored form).
c) h(x) = 5x 4 − 11x 3 + 27x 2 + 125x − 26 (Write answer in factored form).
d) f (x) = 5x 5 + 7x 4 + 170x 3 + 238x 2 − 360x − 504 given that -6i is a zero.
e)
g(x) = 5x 5 + 6x 4 + 30x 3 + 36x 2 − 135x − 162 given that -3i is a zero.