Polynomial and Rational Functions
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Transcript Polynomial and Rational Functions
Polynomial and
Rational Functions
Aim #2.1 What are complex
numbers?
The imaginary unit I is defined as
i 1 where i 1
2
Complex numbers and
Imaginary Numbers
The set of all numbers in the form of a + bi,
with real numbers a and b and i, the
imaginary unit, is called the set of
complex numbers.
An imaginary number in the form of bi is
called a pure imaginary number.
Example: -4 + 6i, 0+ 2i= 2i
A complex number is said to be
simplified if its in the form of a + bi.
If b contain a radical express i before
the radical.
Equality of Complex numbers:
a + bi= c + di
if and only if a = c and b = d
Operations with
Complex Numbers
(5 – 11i) + (7 + 4i)
1.
2.
3.
Steps:
Add or subtract the real
parts
Add or subtract the
imaginary parts
Express final answer as
a complex number
Operations with
Complex Numbers
(-5 + i) – (-11 – 6 i)
1.
2.
3.
Steps:
Add or subtract the real
parts
Add or subtract the
imaginary parts
Express final answer as
a complex number
Multiplying
Complex Numbers
4i(3- 5i)
Distribute 4i throughout
the parenthesis
Multiply
Replace i2 with -1.
Simplify
Multiplying
Complex Numbers
(7 - 3i)(-2- 5i)
Use the Foil Method or
Vertical Method to
multiply
Replace i2 with -1
Simplify
What are Conjugates?
The complex conjugate of the
number a + bi is a – bi and vice
versa.
When you multiply a complex number
by its conjugate you get a real
number.
Using Complex
Conjugates
Divide and express
the result in standard
form.
7 4i
2 5i
Multiply the numerator
and denominator by the
denominators conjugate.
Use FOIL (or the Vertical
Method)
Replace i2 with -1
Simplify
Express final answer in
standard form.
Roots of Negative Numbers
Roots of Negative Numbers
Perform the indicated operation.
a. 18 8
b.(1 5 )
2
25 50
c.
15
Summary:
Answer in complete sentences.
What is i?
Explain how to add or subtract
complex numbers.
What is the conjugate of a complex
number?
Explain how to divide complex
numbers and provide an example.
Aim #2.2: What are some
properties of quadratic functions?
The Standard Form of Quadratic
Function:
f ( x ) a ( x h) k
2
where (h, k ) is the vertex
if a 0 opens upward
if a 0 opens downward
General Form of a Quadratic:
f (x)= ax2 + bx + c
To find the vertex using this form you
need the axis of symmetry:
b
x
2a
Graphing a Quadratic Function
1.
2.
3.
4.
5.
Identify which way the parabola will
open
Identify the vertex
Find the y –intercept by evaluating
f(0).
Find the x-intercepts.
Then graph.
Converting from
General to Standard Form
Convert the function:
Y = x2 + 4x – 1
Steps:
Practice:
Convert to standard form.
Y = 3x2 + 6 x + 7
Summary:
Answer in complete sentences.
3- List three things you learned about
quadratic functions.
2- List 2 ways you can apply this to real
world.
1- Write one question that you may still
have on this topic.
Aim #2.3: How do we identify
polynomial functions and their
graphs?
Examples and Non examples:
Definition of a Polynomial
Function
Smooth, Continuous Graphs
Polynomial functions of degree 2 or
higher have graphs that are smooth
and continuous.
Leading Coefficient Test
Use the leading
coefficient test to
determine the end
behavior.
f (x)= x3 + 3x2 – x - 3
End behavior is how we
describe a graph to the far
left or far right.
Using the Test:
The leading coefficient is 1.
The exponent is odd
Odd-degree have graphs with
opposite behavior at each
end.
Leading Coefficient Test
Odd – degree; positive leading
coefficient
Graphs falls left and increases right
Odd – degree; negative leading
coefficient
Graph rises left and falls right
Guided Practice:
Use the leading
coefficient test to
determine the end
behavior.
f (x)= x4 – 4x2
Even degree; positive
leading coefficient
Rises left and rises right
Even degree; negative
leading coefficient
Falls left and falls right
Using the
Leading
Coefficient Test
Use the leading coefficient test to determine
the end behavior of the graph of:
f (x) = -4x3 ( x -1)2 (x + 5)
Zeros of Polynomial
Functions
Find all the zeros:
f (x)= x3 + 3x2 –x -3
Practice:
Find all the zeros:
f (x)= x3 + 2x2 –4x -8
Finding Zeros of a
Polynomial Function
Find all the zeros:
f (x)= - x4 + 4x3 – 4x2
Steps:
Set f (x) = 0
Multiply both sides by -1.
Factor the GCF.
Factor completely.
Solve for x.
Multiplicities of Zero
Multiplicity and X-intercepts
If r is a zero with
even multiplicity,
then the graph
touches the x-axis
and turns around
at r.
Multiplicity and X-intercepts
If r is a zero with odd multiplicity, then
the graph crosses the x-axis at r.
Note: Regardless of multiplicity
graphs tend to flatten out near the
zeros with multiplicity greater than
one.
Finding Zeros and their
Multiplicities
Find the zeros of f (x)= ½ (x + 1) (2x – 3)2
and give the multiplicity of each zero.
State whether the graph crosses the x-axis
or touches the x-axis and turns around at
each zero.
Steps:
1. Set f (x)= 0 and set each variable factor to 0.
Aim # 2.4 How do we divide
polynomials?
Guided Practice:
The Remainder Theorem
The Factor Theorem
Practice:
Summary:
Answer in complete sentences.
Aim #2.5: How do we find the
zeros of a polynomial function?
Rational Zero Theorem provides us with a
tool we can use to make a list of all
possible rational zeros of a polynomial
function.
Theorem states:
Factors of the cons tan t
Possible rational zeros
Factors of the leading coeffient
Ex 1: Using the Rational Zero
Theorem
Ex. 1 Continued
Now we need to take each number in
the 1st row and divide by each number
in the second row.
How many possible rational zeros are
there?
Practice:
List all the possible rational zeros of:
f ( x) x 2 x 5 x 6
3
2
Ex. 2 Using the Rational Zero
Theorem
Ex. 3 Finding the Zeros of a
Polynomial Function
Part 2
Watch the video: On finding the Zeros
sing the Rational Zero Theorem.
Copy the link into your browser or go to
my web page and click
http://brightstorm.com/math/precalculus/
polynomial-and-rationalfunctions/finding-zeros-of-apolynomial-function//
Summary:
Answer in complete sentences.
Explain how to generate possible
solutions to a polynomial function.
Explain how we use the process of
trial and error and synthetic division to
find the actual zeros of the polynomial
function.
Aim #2.6: How do we find the
asymptotes of a rational function?
Key Terms:
Rational Function
Domain
Vertical Asymptotes
Horizontal Asymptotes
Slant or Oblique Asymptotes
If you were absent from class- watch the videos on my web
page on horizontal and slant asymptotes.
In addition, check a classmates notes.
Summary:
Answer in complete sentences.
What are the different types of
asymptotes?
Explain how to locate the different
types of asymptotes. Be sure to
include examples for each type to
illustrate your explanation.