warm-up: november 16, 2015. homework questions? zeros of polynomial functions section 2.5
DESCRIPTION
Zeros of Polynomial Functions Section 2.5TRANSCRIPT
Warm-Up: November 16, 2015
253125 342 xxxx
Homework Questions?
Zeros of Polynomial Functions
Section 2.5
Roots, Zeros, and -intercepts Functions have zeros Equations have roots Graphs have -intercepts
The following are always the same numbers: The zeros of The roots of The -intercepts of the graph of
Rational Zero Theorem If is a polynomial with integer coefficients,
then every possible rational zero is given by:
Possible rational zeros =
Factors of the constant term Factors of the leading
coefficient
Example 1 (like HW #1-8) List all possible rational zeros for
8526 234 xxxxxf
You-Try #1 (like HW #1-8) List all possible rational zeros for
169734 234 xxxxxf
Finding Zeros of Polynomial Functions1. Use the rational zero theorem to find all the
possible rational zeros.2. Use a guess-and-check method and
synthetic division to find a zero3. Use the successful synthetic division to get a
lower-degree polynomial that can then be solved to find the remainder of the zeros.
Example 3 (like HW #9-22) Find all rational zeros of 41583 23 xxxxf
You-Try #3 (like HW #9-22) Find all rational zeros of 252 23 xxxxf
Example 4 (like HW #9-22) Solve 015162 24 xxx
You-Try #4 (like HW #9-22) Solve 04652 23 xxx
Assignment Read Section 2.5 Page 302 #1-13 odd
An updated 2.5 guided notes has been posted, with a new page 3. It is now 5 pages instead of 4.
You have the rest of class to work on this assignment, or Chapter 1 Test corrections 2.1-2.4 Review
First Four Problems In Exercises 1-8, use the Rational Zero
Theorem to list all possible rational zeros for each given function.
121277)7
6254)5
619113)3
44)1
2345
234
234
23
xxxxxxf
xxxxxf
xxxxxf
xxxxf
Warm-Up: November 17, 2015 Without using your calculator, find all zeros of
12112 23 xxxxf
Homework Questions?
Linear Factorization Theorem An th-degree polynomial has complex roots
Each root can be written as a factor, An th-degree polynomial can be expressed as
the product of a nonzero constant and linear factors: 01
11 ... axaxaxaxf n
nn
n
nn cxcxcxaxf 21
Example 5 (like HW #23-28) Factor the polynomiala) as the product of factors that are irreducible
over the rational numbersb) as the product of factors that are irreducible
over the real numbersc) in completely factored form, including
complex (imaginary) numbers
229 24 xxxf
Example 5 (like HW #23-28) Factor the polynomial
229 24 xxxf
You-Try #5 (like HW #23-28) Factor the polynomiala) as the product of factors that are irreducible
over the rational numbersb) as the product of factors that are irreducible
over the real numbersc) in completely factored form, including
complex (imaginary) numbers
276 24 xxxf
You-Try #5 (like HW #23-28) Factor the polynomial
276 24 xxxf
Assignments Read Section 2.5 Page 302 #1-13 odd, Page 302 #23-27 odd Page 302 #29-35 odd, 43-55 odd (learning
next class) Use your graphing calculator to help find possible
roots for 43-55 odd
You have the rest of class to work on any of the above assignments, Chapter 1 Test corrections, or 2.1-2.4 review.
Warm-Up: November 18/19, 2015 Find all zeros of 152 24 xxxf
Homework Questions?
Finding a Polynomial When Given Zeros1. Write the basic form of a factored
polynomial:
2. Fill in each “” with a zero3. Fractions can be written with the
denominator in front of the “”4. If a complex (imaginary) number is a
zero, so is its complex conjugate5. Multiply the factors together6. Use the given point to find the value of
nn cxcxcxaxf 21
Example 6 (like HW #29-36) Find an degree polynomial function with real
coefficients with the following conditions:
You-Try #6 (like HW #29-36) Find an degree polynomial function with real
coefficients with the following conditions:
Example 7 (like HW #43-56) Find all roots of
024109941124 2345 xxxxx
You-Try #7 (like HW #43-56) Find all zeros of
10611067016 2345 xxxxxxf
Roots, Zeros, and -intercepts Functions have zeros Equations have roots Graphs have -intercepts
The following are always the same numbers: The zeros of The roots of The -intercepts of the graph of
Assignment Read Section 2.5 Page 302 #1-13 odd, Page 302 #23-27 odd Page 302 #29-35 odd, 43-55 odd
Use your graphing calculator to help find possible roots for 43-55 odd