Download In this chapter, you will be able to

Pre-calculus
Name ________________________________________
Review #2 Chapter 3 – Polynomial and Rational Functions
Mrs. Spatola
Pre-Calculus
2010
In this chapter, you will be able to:
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Identify Polynomials and Their Degree
Identify the Zeros of a Polynomial and Their Multiplicity
Analyze the Graph of a Polynomial
Find the Domain of a Rational Function
Determine the Vertical Asymptotes of a Rational Function
Analyze the Graph of a Rational Function
Utilize the Remainder and Factor Theorems
Use Decartes’ Rule of Signs to Determine the Number of Positive and the
Number of Negative Real Zeros of a Polynomial Function
Use the Rational Zeros Theorem to List the Potential Rational Zeros of a
Polynomial Function
Find the Real Zeros of a Polynomial Function
Solve Polynomial Equations
Add, Subtract, Multiply, and Divide Complex Numbers
Solve Quadratic Equations with a Negative Discriminant
Utilize the Conjugate Pairs Theorem to find the Complex Zeros of a Polynomial
Find the Polynomial Function with Specified Zeros
Find the Complex Zeros of a Polynomial
3) Form a polynomial whose zeros and degree are given.
a) Zeros: -3, -5, 5;
b) Zeros: 1, multiplicity 2;
degree 3
zero 5, multiplicity 1;
degree 3
4) For the polynomial, list each real zero and its multiplicity. Determine whether the
graph crosses or touches the x-axis at each x -intercept.
a) f(x) = 4(x - 7)(x - 5)4
b) For the polynomial, list each real zero and its multiplicity. Determine whether the
graph crosses or touches the x-axis at each x -intercept.
f(x) = 5(x2 + 3)(x + 2)2
5) Determine the maximum number of turning points of f.
a) f(x) = 7x - x3
b) f(x) = (x - 2)2(x + 5)2
6) Use the Leading Coefficient Test to describe the right- hand and left- hand behavior
of the graph of each polynomial. (Chapter 3- 3 Lesson 4)
a) f(x) =  x 3  4 x
b) f(x) = x 4  5 x 2  4
c) f(x) = x 5  x
7) Sketch the graph of the polynomial by hand f ( x)  4 x  x 3
a) Find the x & y intercept(s) {only an algebraic solution will be accepted)
b) Using the Leading Coefficient Test, describe the end behavior of the given polynomial
function
c) Determine the maximum number of
turning points f(x) can have.
8) Find the domain of the rational function.
a) F(x) =
b)
c) H(x) =
d)
r(x)=
g(x) =
9) Use the graph to determine the domain and range of the function.
10) Find the vertical asymptotes or holes, if any, of the graph of each rational function.
a) R(x) =
x
x2  4
d) H(x) =
x2
x2 1
b) F(x) =
x3
x 1
c) G(x) =
e) G(x) =
x2  9
x 2  4x  21
11) Asymptotes
– The graph of f has at most one horizontal asymptote determined by comparing the
degrees of p(x) and q(x)
a) If the rational function is proper then y = 0 (the x-axis) is a horizontal asymptote.
b) If the degree of p(x) and q(x) are equal, the graph of f has the line y = an/ bm as a
horizontal asymptote, where an is the leading coefficient of the numerator and bm is the
leading coefficient of the denominator.
c) If the degree of the numerator is exactly one more than the degree of the denominator,
the graph of the function has an oblique (slant) asymptote.
Give the equation of the horizontal asymptote of the graph.
a) R(x) =
c) f(x) =
x  12
4x 2  x  1
x2  x  2
x2  x  6
b) H(x) =
d) P(x) =
13) Give the equation of the oblique asymptote, if any, of the function.
a)
T(x) =
b) H(x) =
c)
Q(x) =
d) f(x) =
2x 2
x2 1
14) Find the
H(x) =
x 2  3x
x2  x  6
a) Domain of H(x)
b) x and y-intercepts, if any, (show algebraically and write your answer as a coordinate pair)
c) vertical asymptote(s) and or hole, if any, write as an equation and/or a coordinate pair
d) horizontal or oblique asymptote if any,
e) Describe the behavior as x  2  f(x)  __________________
Describe the behavior as x  2  f(x)  __________________
x  2
1.9
1.99
1.999
1.9999
[2 points]
F(x)
x  2
F(x)
- 19
- 199
- 1999
- 19999
2.1
2.01
2.001
2.0001
21
201
2001
20001
Name_________________________________
f) Sketch the rational function using all the information in a – e.
15) Given the function f(x) = 2x4 + 5x3 + 20x2 + 45x + 18 find the following.
a)
What is the maximum number of zeros? ___________
b) Using Descartes’ Rule of Signs determine the possible number of positive and negative real zeros.
Positive Zeros______________________________________
Negative Zeros_____________________________________
c) Using the Rational Root Theorem to list all the potential rational zeros.
Potential Zeros:_____________________________________________________________
__________________________________________________________________________
d) Write the polynomial as a product of linear factors
Product of Linear factors:____________________________________________
e) List all complex and real roots.
Real Roots_________________________ Complex
Roots__________________________
16) Solve the equation over the set of complex numbers.
f(x) = x 4  8x 3  64 x  105
a)
What is the maximum number of complex zeros? ___________
b) Using Descartes’ Rule of Signs determine the possible number of positive and
negative real zeros.
c) Using the Rational Root Theorem to list all the potential rational zeros.
d) Find all zeros using a graphing calculator, synthetic division and the quadratic
formula. All work must be shown clearly and neatly.