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Mr. Wolf
Wednesday 10/22/08
Pre-Calculus
Grades 11-12
Unit 2: Polynomial and Rational Functions
Zeros of Polynomial Functions II
Materials and Resources:
 Warm-up (1 per student)
 Zeros of Polynomials Notes II sheet (1 per student)
 Finding Zeros & Writing the Linear Factorization of f(x) sheet (1 per student)
 Finding Zeros & Writing the Linear Factorization of f(x) Example (1 per student)
 Zeros & Linear Factorization Homework sheet (1 per student)
PA Standards Addressed:
2.8.11 S. Analyze properties and relationships of functions (e.g., linear, polynomial,
rational, trigonometric, exponential, logarithmic).
Instructional Objectives:
 Students will be able to find the zeros and linear factorization of functions by
applying the rational root test, Descartes’ Rule, and synthetic division.
Time
10 min
1 min
1 min
25 min
Activity
Warm-up
Agenda
Homework Check
Homework Review
20 min
Rational Zero
Theorem Practice
10 min
Zeros of Polynomials
Notes II
Description
Pass out the warm-up and review solutions.
Review the goals for the day.
Spot check homework sheet.
Review the Zeros of Polynomials Common
Problems Worksheet solutions.
Modeling: Completed when reviewing homework.
Guiding: Help students complete the practice
problems.
Independent Practice: Pg. 179 #18
Assessment: Review solutions
Modifications:
Students with special needs will be given a
textbook to work from.
Advanced students will be called on to offer
solutions.
Modeling: Pass out the notes and provide
information for students to fill in.
Guiding: Help students complete the examples.
Independent Practice: Finding Zeros and Writing
the Linear Factorization of f(x) Example
Assessment: Review solutions.
Modifications:
Students with special needs will be given notes that
are scaffolded to meet their needs.
Advanced students will be instructed to find
solutions to the examples.
20 min
Finding Zeros and
Writing the Linear
Factorization of f(x)
Example
1 min
5 min
Agenda
Conclusion
Modeling: Pass out the notes and example sheets.
Guiding: Help students complete the example
problem.
Independent Practice: Homework exercises
Assessment: Review homework exercises
Modifications:
Students with special needs will be given extra
attention when completing the problem.
Advanced students will be called on to provide
responses during the problem.
Revisit goals and identify whether they were met.
Pass out the Exit Ticket and collect at the bell.
Homework:
Zeros & Linear Factorization Homework sheet
Lesson Reflection:
Pre-Calculus Fall 2008
Name: ________________________
Warm-up
Write the complex number in standard form: 6   28 
Find the difference: (7  8i )  (3  9i ) 
Find the product: (4  3i )(7  2i ) 
Write the quotient in standard form by multiplying by the conjugate of the denominator.
1  8i
 _____ =
7  3i
Pre-Calculus Fall 2008
Name: ________________________
Warm-up
Write the complex number in standard form: 6   28 
Find the difference: (7  8i )  (3  9i ) 
Find the product: (4  3i )(7  2i ) 
Write the quotient in standard form by multiplying by the conjugate of the denominator.
1  8i
 _____ =
7  3i
Pre-Calculus Fall 2008
Name: ________________________
Zeros of Polynomials Notes II
Conjugate Pairs
Example: Given a fourth-degree polynomial f (x) that has -1, 1, and (4 + 3i) as zeros,
what is the fourth zero of the function?
Descartes’ Rule of Signs
Example:
Determine the number of positive and negative real zeros that exist for the function
f ( x)  3 x 3  5 x 2  6 x  4
Positive Zeros:
Negative Zeros:
Pre-Calculus Fall 2008
Name: ________________________
Finding Zeros &Writing the Linear
Factorization of f(x)
Problem: Find the zeros and linear factorization of the function: f (x)
Step 1) Identify the degree and the number of expected zeros of f (x)
degree = greatest exponent of x in f (x)
# of zeros = degree of f (x)
Step 2) Identify the constant term and the leading coefficient term. List all of the
factors of the constant term and put them in a fraction over all of the factors of the
leading coefficient term.
Constant Term = term without x
Leading Coefficient Term = term with greatest exponent of x
Factors of Constant Term =
#’s that multiply to produce the constant term .
Factors of Leading Coefficient = #’s that multiply to produce the leading coefficient term
Step 3) Write all of the possible fractions from Step 2 to create a list of possible
zeros of the polynomial.
Possible Zeros = all fractions that can be created from Step 2
Step 4) Use Descartes’ Rule of Signs to determine the number of positive and
negative zeros.
# of Positive Zeros = # of sign changes in f (x) or less by an even #
# of Negative Zeros = # of sign changes in f ( x) or less by an even #
Based on these numbers, eliminate any possible zeros from Step 3.
Step 5) Find function values by plugging all possible zeros into f (x)
f ( possible _ zero #1) 
f ( possible _ zero #2) 
f ( possible _ zero #3) 
f ( possible _ zero #4) 
Step 6) If any of the function values in Step 5 equal zero, then that value is a zero.
Zeros of f (x) = c = _____
Step 7) Use synthetic division to divide f (x) by all ( x  c ) to obtain other factors.
1st quotient
( x  c1 ) __ f ( x) __
2 nd quotient
( x  c 2 ) 1st quotient
Step 8) Use the divisors (x-c)’s and last quotient from the synthetic division to write
the “initial” factorization of f (x) .
f ( x)  (1st divisor )( 2 nd divisor #2)...(lastdivisior )(lastquotient )
Step 9) Factor any quadratic terms to determine remaining zeros of the polynomial.
Step 10) Write the “final” linear factorization of f (x) .
Pre-Calculus Fall 2008
Name: ________________________
Finding Zeros &Writing the Linear
Factorization of f(x) Example
Problem: Find the zeros and linear factorization of the function: f ( x)  6 x 3  4 x 2  3x  2
Step 1) Identify the degree and the number of expected zeros of f (x)
degree = ________
# of zeros = ________
Step 2) Identify the constant term and the leading coefficient term. List all of the
factors of the constant term and put them in a fraction over all of the factors of the
leading coefficient term.
Constant Term = ________
Leading Coefficient Term = ________
Factors of the Constant Term
=
Factors of the Leading Coefficient
Step 3) Write all of the possible fractions from Step 2 to create a list of possible
zeros of the polynomial.
Possible Zeros =
Step 4) Use Descartes’ Rule of Signs to determine the number of positive and
negative zeros.
# of Positive Zeros
(# of sign changes in f (x) )
=
# of Negative Zeros
=
(# of sign changes in f ( x) )
Based on these numbers, eliminate any possible zeros.
Possible Zeros =
Step 5) Find function values by plugging all possible zeros into f (x)
f (____) 
f (____) 
f (____) 
f (____) 
f (____) 
f (____) 
Step 6) If any of the function values in Step 5 equal zero, then that value is a zero.
Zeros of f (x) = c = _____
Step 7) Use synthetic division to divide f (x) by all ( x  c ) to obtain other factors.
6 x 3  4 x 2  3x  2
Step 8) Use the quotient and divisor from the synthetic division to write the “initial”
factorization of f (x) .
Step 9) Factor any quadratic terms to determine remaining zeros of the polynomial.
Step 10) Write the “final” linear factorization of f (x) .
Pre-Calculus Fall 2008
Name: ________________________
Zeros and Linear Factorization Homework
Find the zeros and linear factorization of the function: f ( x)  2 x 4  6 x 3  8x 2  12 x  8
Step 1) Identify the degree and the number of expected zeros of f (x)
Step 2) Identify the constant term and the leading coefficient term. List all of the factors
of the constant term and put them in a fraction over all of the factors of the leading
coefficient term.
Step 3) Write all of the possible fractions from Step 2 to create a list of possible zeros of
the polynomial.
Step 4) Use Descartes’ Rule of Signs to determine the number of positive and negative
zeros.
Step 5) Find function values by plugging all possible zeros into f (x)
Step 6) If any of the function values in Step 5 equal zero, then that value is a zero.
Step 7) Use synthetic division to divide f (x) by all ( x  c ) to obtain other factors.
Step 8) Use the quotient and divisor from the synthetic division to write the “initial”
factorization of f (x) .
Step 9) Factor any quadratic terms to determine remaining zeros of the polynomial.
Step 10) Write the “final” linear factorization of f (x) .