Download Algebra IIA Unit III: Polynomial Functions Lesson 1

Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior
Leading Coefficient
Polynomial
Polynomial function
o Local Maximum
o Local Minimum
o Synthetic Division
o Monomial
o Multiplicity
o Turning Point
Lesson 1: Polynomials and Multiplying Polynomials
•
•
•
•
Identify, evaluate, add and subtract polynomials (CC.9-12.A.APR.1)
Classify and graph polynomials (CC.9-12.F.IF.7C), (CC.9-12.CED.2 and 3)
Multiply polynomials (CC.9-12.A.APR.1)
Use binomial expansion to expand polynomial expressions raised to positive integer powers (CC.912.A.APR.5)
A constant is ___________________________________________________________________________
A polynomial is _________________________________________________________________________
A monomial is __________________________________________________________________________
A binomial is __________________________________________________________________________
A trinomial is __________________________________________________________________________
The degree of a monomial is found by _______________________________________________________
A leading coefficient is ___________________________________________________________________
A polynomial function is __________________________________________________________________
For the problems above, rewrite each answer in standard form next to the answer you gave on each line.
How do you find the degree of a term with no exponent? _____________________________________
How do you find the degree when there is more than one variable? _____________________________
How can you recognize the leading coefficient? ____________________________________________
For the problems that follow, add or subtract and write your answer in standard form.
1. 3! ! + 7 + ! + 14! ! + 2 + ! ! − !
4. 2! ! + 9 − ! − ! ! − 4 + 5! ! − !
2.
2! ! + 9 − ! + ! ! − 4 + 5! ! − !
5.
9! ! − 4! + 7 + −3! ! − 6! − 5
3.
3! ! + 7 + ! − 14! ! + 2 + ! ! − !
6.
9! ! − 4! + 7 − −3! ! − 6! − 5
7. Why is it a good idea to write polynomials in standard form before adding or subtracting them? _
_____________________________________________________________________________________
_____________________________________________________________________________________
Application Example:
Cardiac output is the amount of blood pumped through the heart. The output is measured by a technique called
dye dilution. A doctor injects dye into a vein near the heart and measures the amount of dye in the arteries over
time. The cardiac output of a particular patient can be approximated by the polynomial function that follows:
! ! = 0.0056! ! − 0.22! ! + 2.33!,where t represents time in seconds after the injection and f(t) represents the
concentration of dye in milligrams per liter.
8. Evaluate f(t) for t = 0 and t = 3.
9. Describe what the values of the function from the previous problem represent.
Multiply the polynomials that follow.
10. 3! ! ! ! + 4
Find each product.
!+! !
!−4
!
11. !" !! + 3!! ! − ! !
12. ! − 2 1 + 3! − ! !
2! − 3
!
3! + 2
!
Assignment: Page 154, 19-45 every other odd, 54-57 Page 162, 27-51 odd, 58-61
Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior√
Leading Coefficient√
Polynomial√
Polynomial function√
o Local Maximum
o Local Minimum
o Synthetic Division
o Monomial√
o Multiplicity
o Turning Point
Lesson 2: Dividing Polynomials
•
Use long division and synthetic division to divide polynomials (CC.9-12.A.APR.2 and 6)
Determine if the statements that follow are sometimes, always, or never true. If you answer ‘sometimes’ please
give an example to justify your answer.
1. A quadratic polynomial is _________________ a trinomial.
2. The degree of a polynomial in standard form is ____________ equal to the degree of the first term.
3. The leading coefficient of a polynomial is _______________ the greatest coefficient of any term.
P(x) and R(x) are polynomials. P(x) is a trinomial. Give examples of P(x) and R(x) that meet the given
conditions.
4. ! ! − ! ! is a binomial------------------------ ! ! = ____________________________ and ! ! = ___________
5. ! ! − ! ! is a trinomial------------------------ ! ! = ____________________________ and ! ! = ___________
6. ! ! − ! ! is a polynomial with four terms.- ! ! = ____________________________ and ! ! = ___________
7. ! ! − ! ! is a fourth degree polynomial. --- ! ! = ____________________________ and ! ! = ___________
Divide the following without a calculator. (Long Division)
!""
!"#
!! ! !!!!!
8. !"
9. !
10.
!!!
11.
!! ! !!! ! !!
!!!
Synthetic Division
Use synthetic division to divide the polynomials.
12.
!! ! !!"!!!
!
!!
!
13.
!! ! !!! ! !!"
!!!
Use synthetic division to evaluate the polynomial.
15. ! ! = 2! ! + 5! ! − ! + 7, ! = 2
14.
!! ! !! ! !!!!!
!!!
!
16. ! ! = 6! ! − 25! ! − 3! + 5, ! = − !
Assignment: Page 170, 13-33 odd, 53-55
Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior√
Leading Coefficient√
Polynomial√
Polynomial function√
o Local Maximum
o Local Minimum
o Synthetic Division√
o Monomial√
o Multiplicity
o Turning Point
Lesson 3: Factoring Polynomials
•
•
Use the Factor Theorem to determine factors of a polynomial (CC.9-12.A.APR.2)
Factor the sum and difference of two cubes (CC.9-12.A.APR.2)
Warm-Up
Factor each expression.
1. 3! − 6!
2. !! + ! !
Find the product.
3. ! − 1 ! + 3
4.
! − 1 !! + 1
Expand each expression.
5. 2! − 3 !
6.
3!! ! − 2!
!
Factor the polynomials.
7.
! ! = ! ! + 3! ! − 4! − 12
8.
! ! − 2! ! − 9! + 18
9.
! ! = 2! ! + ! ! + 8! + 4
Rational Zeros Theorem
The Rational Zero Theorem is used to determine possible
factors of a polynomial function.
Determine whether the given binomial is a factor of the polynomial.
10. ! − 5 ; ! ! = ! ! + 2! − 3
14. ! + 2 ; ! ! = 4! ! − 2! + 5
11. ! − 3 ; ! ! = ! ! + 2! − 3
15. 3! − 6 ; ! ! = 3! ! − 6! ! + 6! ! + 3! − 30
12. ! − 1 ; ! ! = ! ! + 2! − 3
16. ! + 2 ; ! ! = 3! ! + 6! ! − 5! − 10
13. ! + 4 ; ! ! = 2! ! + 8! ! + 2! + 8
17. ! + 1 ; ! ! = ! ! − 3! + 1
Factor each expression completely.
18. 5! ! + 40!
19. 8! ! − 27
20. 25! ! + 40!
21. 64 + 8! !
22. 10! ! − 1000! !
23. 4! ! + 108!
24. 125! ! − 8! !
Factor.
25. ! ! − 5! ! − ! + 5
31. 2! ! − 13! + 20
26. ! ! − 2! ! − 15
32. 4! ! − 61! ! + 225
27. ! ! − 26! ! − 27
33. 5! ! + 24! − 5
28. ! ! + 2! ! − 16! ! − 32
34. 5! ! + 29! + 20
29. ! ! − 5! ! − ! + 5
35. 4! ! + 4! − 15
30. 2! ! + ! ! − 6
36. 10! ! − 8! ! + 25! − 20
Assignment: Page 177, 19-47 odd, 50, 51
Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior√
Leading Coefficient√
Polynomial√
Polynomial function√
o Local Maximum
o Local Minimum
o Synthetic Division√
o Monomial√
o Multiplicity
o Turning Point
Lesson 4: Finding Real Roots of Polynomials and the Fundamental Theorem of Algebra
•
•
•
•
Identify the multiplicity of roots. (CC.9-12.A.APR.3)
Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations. (CC.912.A.REI.11) and (CC.9-12.A.CED.1)
Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree
with given roots. (CC.9-12.N.CN.9)
Identify all of the roots of a polynomial equation. (CC.9-12.N.CN.7,9) and (CC.9-12.A.APR.2)
Warm-Up
Draw a parabola in the first grid, a ‘curvy’ cubic in the second grid and a ‘curvy’ W in the third grid. Now,
after your graphs are drawn, draw an x-axis and a y-axis wherever you like.
1. # of zeros: ___________
4. # of zeros: ___________
2. # of zeros if the x-axis
moved down 5 units:__
3. # of zeros if the x-axis
moved up 5 units:___
5. # of zeros if the x-axis
moved down 5 units:___
6. # of zeros if the x-axis
moved up 5 units: _____
7. # of zeros: ___________
8. # of zeros if the x-axis
moved down 5 units:___
9. # of zeros if the x-axis
moved up 5 units:____
Solve each polynomial by factoring and then use a graphing calculator to help graph each function in the grid
provided.
10. 2! ! − 10! ! − 12! ! = 0
11. ! ! − 2! ! − 25! = −50
12. 3! ! + 18! ! + 27! ! = 0
The multiplicity of a root ___________________________________________________________________________________
________________________________________________________________________________________________________
________________________________________________________________________________________________________
Multiplicity
Irrational Zeros Theorem
The Irrational Zero Theorem
states that if a polynomial
function has rational
coefficients and ! + ! ! is a
root, then ! − ! ! is also a root.
If __________ is a root, then
________ is also a root.
Rational Zeros Theorem
The Rational Zero Theorem is used to determine possible
factors of
a polynomial function.
The Fundamental Theorem of
Algebra
Any polynomial of degree n has
exactly n solutions (could be
real or complex in nature).
If the polynomial is
______________
then it has
____________solutions.
The Complex Conjugate Root
Theorem
If ! + !" is a root of a
polynomial equation with real
number coefficients, then
! − !" is also a root of the
polynomial equation.
If __________ is a root, then
________ is also a root.
Determine the roots of each equation. State the multiplicity of each root.
13. ! ! − 9! ! + 27! − 27 = 0
14. ! ! − 8! ! + 24! ! − 32! + 16 = 0
16. A popcorn producer is designing a new box
for the popcorn. The marketing department
has designed a box with the width 2 inches
less than the length and with the height 5
inches greater than the length. The volume
of each box must be 24 cubic inches. What
is the length of the box?
15. 2! ! − 22! ! + 48! ! + 72! ! = 0
17. A shipping crate must hold 12 cubic feet.
The width should be 1 foot less than the
length and the height should be 4 feet
greater than the length. What should the
length of the crate be?
Identify the roots of the polynomial equations that follow. (Directions could also be: solve the equation or find
the zeros.)
18. ! = 2! ! − 3! ! − 10! − 4
19.
4! ! − 21! ! + 18! ! + 19! − 6 = ! !
20. ℎ(!) = 2! ! − 9! ! + 2
Identify the roots of the polynomial equations that follow. (Directions could also be: solve the equation or find
the zeros.)
21. ! ! + ! ! + 2! ! + 4! − 8 = !
22. ! ! = ! ! + 4! ! − ! ! +
16! − 20
23. ! ! + 13! − 85 = 31
Write the simplest polynomial function given the following zeros. (Directions could also be: write a
polynomial with integral coefficients.)
!
26. 0, ! , 3
28. 3 + 5
!
27. ±2!
29. 2, !
24. −3, ! , 1
!
25. ! , − !
!
Assignment: Page 186, 15-33 odd, 40-43, page 193, 21-43 odd, 57-61
Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior√
Leading Coefficient√
Polynomial√
Polynomial function√
o Local Maximum√
o Local Minimum√
o Synthetic Division√
o Monomial√
o Multiplicity√
o Turning Point√
Lesson 5: Investigating Graphs of Polynomial Functions and Transforming Polynomial
Functions
•
•
•
Use properties of end behavior to analyze, describe, and graph polynomial functions. (CC.9-12.A.APR.3)
and (CC.9-12.F.IF.7c)
Identify and use maxima and minima of polynomial functions to solve problems. (CC.9-12.A.CED.2,3)
Transform polynomial functions. (CC.9-12.F.IF.7c), (CC.9-12.F.BF.3), and (CC.9-12.A.CED.2,3)
Warm-Up:
Determine whether each statement is sometimes, always, or never true. If it is sometimes true, give examples to
support your answer. The letters a, b, and c represent real numbers.
1. A cubic polynomial has no real zeros. _______________________________________________________
2. A quartic polynomial has an odd number of real zeros. __________________________________________
3. There are infinitely many polynomials with zeros a, b, and c. _____________________________________
4. The multiplicity of a root is equal to the degree of the polynomial. _________________________________
5. The volume of a giant sequoia can be modeled by ! ℎ = 0.485ℎ! − 362ℎ! + 89889ℎ − 7379874, where
h represents the height of a tree in feet and 220 ≤ ℎ ≤ 280.
a. Find the height of a tree that has a volume 39,186 cubic feet.
b. The Lincoln tree in the Giant Forest in Sequoia National Park has a volume of 44,471 cubic feet.
What are the possible heights of the tree?
c. The actual height of the Lincoln tree is 255.6 feet. What is the difference between the true volume
of the tree and the volume given by the model?
The Leading Coefficient Test
Given a polynomial function in standard form
+ ⋯ + !! ! + !! :
There are four cases that go with this test:
!
! ! = !! ! + !!!! !
!!!
Case 1: If n is odd AND the leading coefficient !! , is positive,
the graph falls to the left and rises to the right:
! ! → −∞ !" ! → −∞ !"# ! ! → ∞ !" ! → ∞.
Case 3: If n is even AND the leading coefficient !! ,
is positive, the graph rises to the left and to the right.
Case 2: If n is odd AND the leading coefficient !! , is
negative, the graph rises to the left and falls to the right.
Case 4: If n is even AND the leading coefficient !! , is
negative, the graph falls to the left and to the right.
Use the Leading Coefficient Test to determine the end behavior of the graph of ! ! = 5! ! − 2! ! + 5.
Think:
What is the leading term?
What is the leading term’s degree?
What is the coefficient on the leading term?
Examining Graphs
6.
7.
8.
9.
Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
10. Degree of polynomial: ____
11. Number of zeros: ________
12. Number of turns: ________
13. End Behavior: __________
14. Degree of polynomial: ____
15. Number of zeros: ________
16. Number of turns: ________
17. End Behavior: __________
By inspecting the equation, identify the attributes of the following functions and then graph the polynomials
with a graphing calculator to check your answers.
18. ! ! = ! ! − !
Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
19. ! ! = ! ! − 8! ! + 1
Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
20. ! ! = 3! ! − 2! + 2
Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
Write an equation for the functions with the following attributes.
21. Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
Function: ______________
22. Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
Function: ______________
Steps for Graphing a Polynomial Function
1. Find the zeros (x-intercepts) and yintercepts.
2. Plot the intercepts.
3. Make a table of x-values that lie between the
zeros.
4. Plot the points from the table.
5. Determine the end behavior of the graph.
6. Sketch the graph.
Graph the functions that follow.
24. ! ! = ! ! + 3! ! − 6! − 8
23. Degree of polynomial: ____
Number of zeros: ________
Number of turns: ________
End Behavior: __________
Function: ______________
Anatomy of a Polynomial Function
25. ! ! = −2! ! − ! + 6
Function Transformations…
Write the rule for each function. Use your graphing calculator to confirm that your answer is correct.
26. ! ! = ! ! − 8 translated up four units. ____________________________________________________
27. ! ! = ! ! − 8 translated right five units.___________________________________________________
28. ! ! = ! ! − 8 translated down 7 units._____________________________________________________
29. ! ! = ! ! − 8, ! ! = −! ! . ___________________________________________________________
30. ! ! = ! ! − 8, ! ! = ! ! + 3 . _________________________________________________________
31. ! ! = ! ! − 8, ! ! = ! −! . ___________________________________________________________
Write the function that transforms ! ! = 8! ! − 2 in each of the following ways. Use your graphing calculator
to confirm that your answer is correct.
32. Compress the function vertically by a factor of one half and move the x-intercept to the right three units.
33. Reflect the function across the x-axis and move the function four units to the left.
Assignment: Page 201, 23-41 odd, 48, 49, Page 207, 15-29 odd, 31-33.
Algebra IIA
Unit III: Polynomial Functions
•
Foundational Material
•
Goal
•
Why?
•
Key Vocabulary
o Use transformations to graph linear and quadratic
functions
o Solving linear and quadratic equations and
inequalities
o Graph and transform
polynomial functions
o Fit data using linear and quadratic models
o Use and perform operations with real, imaginary
and complex numbers
o Solve polynomial equations
and inequalities
o Fit data to polynomial models
o Build a foundation for higher level mathematics
o These skills can be used to observe, understand, and model relationships in science, social studies and economics
o Maximize and minimize area and volume
o
o
o
o
End behavior√
Leading Coefficient√
Polynomial√
Polynomial function√
o Local Maximum√
o Local Minimum√
o Synthetic Division√
o Monomial√
o Multiplicity√
o Turning Point√
Lesson 6: Curve Fitting with Polynomial Models
•
•
Use finite differences to determine the degree of a polynomial that will fit a given set of data. (CC.912.F.IF.7c), and (CC.9-12.A.CED.3)
Use technology to find polynomial models for a given data set. (CC.9-12.A.CED.2)
How to determine the degree of a polynomial from data.
1.
Use finite differences to determine the degree of the polynomial that would be modeled by the data that follows.
x
y
-2
-10
-1
-4
0
-1.4
1
0
2
2.4
0
1
2
3
8
The x-values increase by a constant of 1.
x
Let’s consider the y-values.
-10
-4
-2
-1
-1.4
0
2.4
3
8
The third differences are constant. This data would be best modeled by a cubic function.
2.
Use finite differences to determine the degree of the polynomial that would be modeled by the data that follows.
x
y
-6
-30
-4
15
-2
30
-4
-2
0
34
2
41
0
2
4
60
The x-values increase by a constant of 2.
x
Let’s consider the y-values.
-30
-6
15
30
34
41
4
60
The fourth differences are constant. This data would be best modeled by a quartic function.
3.
Use finite differences to determine the degree of the polynomial that would be modeled by the data that follows.
x
y
12
3
15
23
18
29
21
29
24
31
27
43
The ___________ differences are constant. This data would be best modeled by a __________ function.
Why would we want to know this? ______________________________________________________________________
___________________________________________________________________________________________________
___________________________________________________________________________________________________
(If you aren’t sure of how to answer this question, try the next problem.)
The table below shows the gas consumption of a compact car driven a constant distance at various speeds. Use your calculator to write a
regression function for this data.
Speed
Gas (gallon)
25
23.8
30
25
35
25.2
40
25
45
25.4
50
27
55
30.6
60
37
The table below shows the opening value of a stock on the first day of trading in various years. Use a polynomial model to estimate the
value on the first day of trading in 2000.
Year
1994
1995
1996
Price ($)
683
652
948
Year
1997
1998
1999
Price ($)
1306
863
901
Assignment: Page 213, 6-12, 17-18, Page 217, Ready To Go On 1-18