Upload
holli
View
45
Download
3
Tags:
Embed Size (px)
DESCRIPTION
Chapter 2 Power, Polynomial, and Rational Functions. 2-4 Zeros of a Polynomial Function. Warm-up . Factor using long division. 1. Find f(c) using synthetic substitution. 2. Homework Check. …and a short Homework Quiz . Complete the following:. Recall the values of the following: - PowerPoint PPT Presentation
Citation preview
2-4Zeros of a Polynomial Function
Chapter 2Power, Polynomial, and Rational Functions
Warm-up
Factor using long division.1.
Find f(c) using synthetic substitution.2.
3 ;968204 23 xxxx
4 ;382423)( 3456 cxxxxxxf
Homework Check
…and a short Homework Quiz
Complete the following:
Complex Numbers(2 – 3i, 2i, 16, )
Recall the values of the following:
i = i2 = i3 = i4 = i5 = i6 = i7 =
•Fractions•Integers•Irrational numbers•Imaginary numbers• Natural numbers•Rational numbers•Real numbers•Whole numbers
1
Complex Numbers(2 – 3i, 2i, 16, )
Real Numbers
Rational Numbers
Fractions Integers
Whole Numbers
Natural Numbers
Irrational Numbers
Imaginary
Numbers
Recall the values of the following:
i = i2 = i3 = i4 = i5 = i6 = i7 =
•Integers•Irrational numbers•Imaginary numbers• Natural numbers•Rational numbers•Real numbers•Whole numbers
1
Objectives for 2-4
Find real zeros of polynomial functions Find complex zeros of polynomial
functions
1. Real Zeros of a Polynomial The leading coefficient and constant
term with integer coefficients can be used to determine a list of all possible rational zeros.
Then you can determine actual zeros using synthetic division.
This is the Rational Zero Theorem
Rational Zero TheoremEvery rational zero of a polynomial has the
form , where p is an integer factor of the constant
term q is an integer factor of the leading
coefficient
qp
Example 1: List all possible rational zeros. Then determine which, if any, are zeros.
423)( 23 xxxxf423)( 23 xxxxf
Example 2: List all possible rational zeros. Then determine which, if any, are zeros.
12)( 3 xxxf
Example 3: List all possible rational zeros. Then determine which, if any, are zeros.
152852)( 3 xxxxf152852)( 3 xxxxf
2. Writing a Polynomial Given its Zeros
Write a polynomial function of least degree with real coefficients in standard form that has -1, 2, and 2 – i as zeros.
That’s enough for one day… Practice these skills, and then we will
put everything together after the mid-chapter quiz next class.
The Mid-chapter quiz on Tuesday…Topics covered (2 – 1 through 2 – 3): Domain and Range of graphed functions Solving radical equations Determining end behavior of a polynomial
without the use of a calculator Determining the number of turning points
and where functions increase and decrease
Using long division, synthetic division, and synthetic substitution to determine factors of polynomial functions
Assignment due Thursday Practice with skills from today’s
lessonp. 127, #3, 5, 11, 13, 15, 33, 35, 37.
Finish the Mid-chapter quiz
Data analysis
38 48 58 68 78 88 98 10850
60
70
80
90
100
f(x) = 0.391705456620319 x + 46.4127211027358R² = 0.481288947669606
Quarter 1 Average
Homework Grade
Qua
rter
1 A
vera
ge
Almost 14% of the variability in overall grade can be attributed to the homework grade.
38 48 58 68 78 88 98 10850
60
70
80
90
100
f(x) = 0.176381049945879 x + 63.9334425416216R² = 0.135776830314401
Quarter 1 Average
Homework Grade
Qua
rter
1 A
vera
ge
Almost half of the variability in overall grade can be attributed to your homework average.
38 48 58 68 78 88 98 10850
60
70
80
90
100
f(x) = 0.391705456620319 x + 46.4127211027358R² = 0.481288947669606
Quarter 1 Average
Homework Grade
Qua
rter
1 A
vera
ge
Homework Check:
A couple more concepts will be of some help in pulling all the ideas together in this section.
Upper and Lower Bounds Tests (EASY!)
Descartes’ Rule of Signs (non-essential)
Pull it all together with the first half of this lesson
Using the Upper and Lower Bounds Test To narrow the search for real zeros,
you can determine the interval in which the real zeros are located.
This function seems have real zeros between -2 and 2. Eliminate all zeros outside that interval.How easy is that!?
Lower Bound Upper Bound
Descartes’ Rule of Signs tells you the number of positive or negative real zeros
If you are interested, you can find this on p. 123 in your textbook.
It is non-essential, but an interesting theoretical construct by the great French mathematician Decartes.
Pulling it all together!
Factor and find the zeros (both real and irreducible quadratic factors)
Then, factor the irreducible quadratic factors into imaginary roots and list all the zeros.
Example:
Write k(x) as the product of linear and irreducible quadratic factors.
Write k(x) as the product of linear factors.
List all the zeros of k(x).
24142313)( 2345 xxxxxxk
Example:Write k(x) as the product of linear and irreducible quadratic factors.
Step 1: List all possible factors 24142313)( 2345 xxxxxxk
Example:Write k(x) as the product of linear and irreducible quadratic factors.
Step 2 (optional) Check Descartes Rule of Signs
24142313)( 2345 xxxxxxk
Example:Write k(x) as the product of linear and irreducible quadratic factors.
Step 3: Look at the graph in your calculator and find upper and lower bounds of the real roots.
Eliminate all possible roots outside of the upper and lower bounds.
Start testing with those.
24142313)( 2345 xxxxxxk
Example:Write k(x) as the product of linear and irreducible quadratic factors.
The graph suggests that 4 is a zero. Start there.
Use the depressed polynomial to test the next possible zero.
24142313)( 2345 xxxxxxk
Example:Write k(x) as the product of linear and irreducible quadratic factors.
The graph suggests that -2 is another zero. Try that one.
Use the depressed polynomial to test the next possible zero.
6575 234 xxxx
Example:Write k(x) as the product of linear and irreducible quadratic factors.
The graph suggests that -3 is another zero. Try that one.
Write the depressed polynomial. Note that it is irreducible (It can’t be factored with real roots).
33 23 xxx
Summarize all the roots and factors Roots: 4, -2, and -3
Factors:
Assignment: p. 127, 39 – 47 odds