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Warm-Up: January 9, 2012
253125 342 xxxx
Homework Questions?
Zeros of Polynomial Functions
Section 2.5
Rational Zero Theorem If f(x) 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
Warm-Up: January 10, 2012 Find all rational zeros of
12112 23 xxxxf
Homework Questions?
Example 4 (like HW #9-22) Solve 015162 24 xxx
You-Try #4 (like HW #9-22) Solve 04652 23 xxx
Linear Factorization Theorem An nth-degree polynomial has n complex roots
c1, c2, …, cn
Each root can be written as a factor, (x-ci) An nth-degree polynomial can be expressed as
the product of a nonzero constant and n 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
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
Warm-Up: January 11, 2012 Find all zeros of
Hint: Start by factoring as we did in Example 5.
152 24 xxxf
Homework Questions?
Finding a Polynomial When Given Zeros1. Write the basic form of a factored
polynomial:
2. Fill in each “c” with a zero3. Fractions can be written with the
denominator in front of the “x”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 an
nn cxcxcxaxf 21
Example 6 (like HW #29-36) Find an nth degree polynomial function with
real coefficients with the following conditions: n = 4 Zeros = {-2, -1/2, i} f(1) = 18
You-Try #6 (like HW #29-36) Find an nth degree polynomial function with
real coefficients with the following conditions: n = 3 Zeros = 4, 2i f(-1) = -50
Warm-Up: January 12, 2012 Simplify and write in standard form (refer to
2.1 notes if needed)
i
i
35
32
Homework Questions?
Descarte’s Rule of Signs Let f(x) be a polynomial with real coefficients1. The number of positive real zeros of f is
either equal to the number of sign changes of f(x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero.
2. The number of negative real zeros of f is either equal to the number of sign changes of f(-x) or is less than that number by an even integer. If f(-x) has only one variation in sign, there is exactly one negative real zero.
Example 7 (like HW #43-56) Find all roots of
024109941124 2345 xxxxx
You-Try #7 (like HW #43-56) Find all zeros of
102112 23 xxxxf
Assignment Complete one of the following assignments: Page 302 #1-33 Every Other Odd, 43 OR Page 302 #43-55 Odd
Chapter 2 Test next week You may use a 3”x5” index card (both sides)