Pre-calculus Mrs. Spatola
Name ______Pre-Calculus
Review #2 Chapter 3 – Polynomial and Rational Functions
In this chapter, you will be able to:
§ 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
1) State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
a) f(x) = 5x + 6x4
b) f(x) =
c) f(x) =
d) f(x) = 6(x - 1)11(x + 1)5
2) Determine which of the following are polynomial functions.
a) g(x) = b) h(x) =
c) f(x) = 0 d) f(x) = 8
d) e) f(x) = 2 – 3x
3) Form a polynomial whose zeros and degree are given.
a) Zeros: -3, -5, 5; degree 3
b) Zeros: 1, multiplicity 2; 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) Find the power function that the graph of f resembles for large values of |x|.
a) f(x) = -x2(x + 9)3(x2 - 1)
b) f(x) = 3x - x3
6) Determine the maximum number of turning points of f.
a) f(x) = 7x - x3 b) f(x) = (x - 2)2(x + 5)2
7) 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) = b) f(x) =
c) f(x) =
8) Sketch the graph of the polynomial by hand
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.
9) Find the domain of the rational function.
a) F(x) = b) r(x)=
c) H(x) = d) g(x) =
10) Use the graph to determine the domain and range of the function.
11) Find the vertical asymptotes or holes, if any, of the graph of each rational function.
a) R(x) = b) F(x) = c) G(x) =
d) H(x) = e) G(x) =
12) 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) = b) H(x) =
c) f(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) =
14) Find the H(x) =
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 f(x) ______[2 points]
Describe the behavior as f(x) ______
/ F(x) / / F(x)
1.9 / - 19 / 2.1 / 21
1.99 / - 199 / 2.01 / 201
1.999 / - 1999 / 2.001 / 2001
1.9999 / - 19999 / 2.0001 / 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.
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) =
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.