# Matrices

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```					8-1.6 Factoring Polynomials                                                               TEACHER INSTRUCTIONS

ALGEBRA
2. Use algebraic concepts to identify patterns, use multiple representations of relations and functions, and apply
operations to expressions, equations, and inequalities.
f. Factor sums and differences of cubes and factor polynomials by grouping. (DOK 2)
l. Interpret the zeros and maximum or minimum value(s) of quadratic functions. (DOK 2)

Source: FactoringCAS & Algebra 2 Week 16 “One of the Many Ways”, from TI Activities Exchange

Materials:
TI-Navigator System
TI-84 calculators
Files: Day 1: BR 8.1.edc, Student handout - 8.1&2 Lesson, 8.1 & 8.2 How Long Is the Soccer Field.flp
Day 2: BR 8.2.edc, HW Quiz 8A.edc, Student handout - 8.1&2 Lesson, 8.1 & 8.2 How Long Is the Soccer Field.flp
Day 3: BR 8.3.edc, HW Quiz 8B.edc, Student handout - 8.3 Lesson, 8.3 Can You Carry-on Your Luggage.flp
Day 4: BR 8.4.edc, HW Quiz 8C.edc, Cards for Factoring Sort – See Factoring Sort.pdf, Factoring_Jeopardy.ppt,
FactoringJeopardy.ppt, factoring-jeopardy.html
Day 5: BR 8.5.edc, HW Quiz 8D.edc, Unit 8 Review.doc, Unit 8 Review.flp, Review in Class.flp
Day 6: BR 8.6.edc, Unit 8 Assessment.doc

Unit 8 Objectives.doc is provided so that you can print out the objectives to post in your room.

This unit will take 3 days to guide students through the flipcharts. The fourth day will be a quiz and review. The
fifth day will be another quiz and more review. The sixth day will be an assessment.
th
On the 4 day, see Factoring Sort.pdf for ideas about how to use the cards. Also spend some time using Activity
Center so that students are held accountable for practicing factoring.

There are several Jeopardy Powerpoints in your folder: Factoring_Jeopardy.ppt, FactoringJeopardy.ppt, and an
html link to one that students can play online. Use as needed.

Use Review in Class.flp with your class as you see fit. It is an extra flipchart with multiple choice practice.

Use Activity Center throughout these lessons. Load Submit an Equation.act so that students can submit an
equation. You can enter an equation in standard form & have the students send you the factored form.
Emphasize that the graphs must be the same if the two forms are equivalent.
8.1 & 8.2 How Long Is the Soccer Field?                                 Name__________________________________

I. The optimum area for a soccer field is 9000 square yards. If the width of the field is (2x+5) yd and its length is
(3x+15) yd, then what are the dimensions of the field?

GCF Skills Needed!
A. 3a2  9                B. 25b2  35             C. x2  2 x                D. 5t 2  7t            E. 14 y 2  7 y

II. Recognizing the Difference of Squares
1. When compared to its parent, what happens to the graph of y  x 2  1 ?
How many times does it cross the x-axis, and where?
What is the factored form of y  x 2  1 ?

Your instructor will use the TI-Nspire CAS to factor several other binomials, so that you can recognize the pattern.
2. What is the factored form of y  x2  2 ?
3. Factor y  x2  3 .
4. Factor y  x2  4 .
5. Over what set of numbers are you factoring?
6. What binomial should factor next?
What is its factored form?
7. These binomials are called perfect square binomials, and they factor as the difference of squares. List three
other perfect square binomials, and write the factored form of each.

8. What is the factored form of x2  a2 ?

III. Recognizing the Sum of Squares
9. When compared to its parent, what happens to the graph of y  x 2  1 ?
How many times does it cross the x-axis, and where?
What is the factored form of y  x 2  1 ?

10. What is the factored form of x2  a2 over the set of rational numbers?

IV. Binomial Practice: Rewrite each in factored form.

11. 4 x 2  25 y 2        12. 4 x2  36            13. 121y 6  9 x 2         14. x4  16             15. x2  100

V. Recognizing Patterns for Trinomials ++
16. How many times does y  x 2  13x  36 cross the x-axis, and where?
What is the factored form of
y  x 2  13x  36 ?

Factor each of the following
trinomials.

17. y  x 2  14 x  48

18. y  x 2  12 x  27
Create the trinomial represented
by each X, and write its factored
form.

19.

20.

21.

22.

23.

24.

In your words – How do you factor a polynomial in the form x2  bx  c ?

VI. Recognizing Patterns for Trinomials -+
25. How many times does y  x 2  13x  36 cross the x-axis, and where?
What is the factored form of y  x 2  13x  36 ?

Factor each of the following trinomials.
26. y  x 2  14 x  48                  27. y  x 2  12 x  27           28. y  x 2  9 x  20

In your words – How do you factor a polynomial in the form x2  bx  c ?

VII. Recognizing Patterns for Trinomials +- and --
29 . How many times does y  x 2  5 x  24 cross the x-axis, and where?
What is the factored form of
y  x 2  5 x  24 ?

Factor each of the following
trinomials.

30. x2 14x  95

31. x2  4 x  60

Create the trinomial represented by
each X, and write its factored form.

32.

33.

34.
35.

36.

37.

In your words – How do you factor a polynomial in the form x2  bx  c ?

In your words – How do you factor a polynomial in the form x2  bx  c ?

VIII. Recognizing Patterns for Perfect Square Trinomials
38. When compared to its parent, what happens to the graph of y   x  3 ?
2

How many times does it cross the x-axis, and where?
What is the expanded form of y   x  3 ?
2

39. When compared to its parent, what happens to the graph of y   x  4  ?
2

How many times does it cross the x-axis, and where?
What is the expanded form of y   x  4  ?
2

Expand each of the following.
40.  x  8               41.  x  9               42.  2 x  5            43.  3 x  2               44.  4x  y 
2                          2                            2                         2                           2

In your words – How do you expand a binomial in the form  x  a  ?
2

Factor each of the following perfect square trinomials.
45. x2  2 x  1         46. x2  6x  9          47. x2  8x  16              48. 4 x 2  20 xy  25 y 2   49. x2 12x  36

In your words – How do you recognize a perfect square trinomial?

IX. Recognizing Patterns for Factoring the Difference of Cubes
50. When compared to its parent, what happens to the graph of y  x3  1 ?
How many times does it cross the x-axis, and where?
Your instructor will use the TI-Nspire CAS to factor cubics, so that you can recognize the pattern.
What is the factored form of y  x3  1 ?

Multiply the factored form to verify that it is correct.

Write the factored form of each of the following. Look for patterns.
51. x3  8                                 52. x3  27                                    53. x3  64

54. x3  125                              55. x3  343                                    56. x3  a3
X. Factoring the Sum of Cubes
57. x3  1                                   58. x3  27                                59. x3  64

60. What is the factored form of x3  a3 ?

XI. Cubic Practice: Rewrite each in factored form.
61. 8x3  y3                              62. p3  27q3                                 63. 64 x3  27 y 3

XII. Factoring by Grouping
64. x3  3x2  4x  12                                     65. x3  x2  2x  2

XIII. Factoring Applications
66. The area of a fenced-in garden must be 432 square feet to accommodate a water feature. What should the
dimensions of the garden be if they measure (x) ft by (42-x) ft?

67. A rectangular swimming pool has a volume of 512 cubic feet. The pool’s dimensions are (x) ft deep by (6x-8)
feet long by (6x-16) feet wide. How deep is the pool?

XIV. Factoring Practice
1. x2  9x  14                   2. 2x2  13x  6                  3. x2  12x  36                         4. x2  100

5. 2x6 10x4 12x2                6. 64x2  32 x  4                7. 16x2  40x  25                       8. 9 x 2  16

9. 3x2  9 x  6                  10. x3  216                      11. 6x2  48x  54                       12. 12 x 2  3

13. 64x3  y3                     14. 49x2  42 x  9               15. 54 y 4  16 y                        16. 16 x 2  4

17. 16x4 1                       18. x4  7 x2  12                19. x5  x3  2 x                        20. x3  8 y3

21. 2 x2  8x  2                 22. x2  8x  16                  23. 81x2 1                              24. x3  1000

2 x  10                        x 1                              x 2  5 x  6 x 2  3x  2               6x2  9x
25.                               26.                               27.                 2                   28.
x  10 x  25
2
x2  1                                x2  4    x  2x  3                   3x

6x  6 y   3x                     x 2  8 x  16                    2 x  12 2 x  6                            2x
29.             2                30.                               31.                                     32.
x y     x 1                    x 2  2 x  24                    3x  9 3 x  8                           4 x2  2 x
8.3 Can You Carry-on Your Luggage                                                Name__________________________________

I. Several popular models of carry-on luggage have a length 10 in. greater than their depth. To comply with airline
regulations, the sum of the length, width, and depth may not exceed 40 in.
a. Assume that the sum of the length, width, and depth is 40 in. Graph the function relating volume V to depth x.
Find the x-intercepts. What do they represent?
b. Describe a realistic domain.
c. What is the maximum possible volume of a piece of luggage? What are the corresponding dimensions of the
luggage?

II. Making Connections
A. Write a polynomial in standard form with the given zeros.
1. x  2,0,1                         2. x  5M2,1                                            3. x  3M3

B. Use a graphing calculator to find the relative maximum, relative minimum,and zeros of each function.
4. f  x   x3  4 x 2  5x           5. f  x    x3  16 x 2  76 x  96 6. f  x   x 4  3x3  x 2  3x

C. Factoring Functions
7. f  x   3x3  27 x 2  24 x                  8. f  x   2 x3  2 x 2  40 x            9. f  x   x 4  3x3  4 x 2

D. Comparing Graphs
10. What transformation could you use to describe the change from the graph of y   x  1 x  2  x  3 to the
graph of y   x  1 x  2  x  3 ?

11. Does the same transformation describe the change from the graph of y   x  1 x  3 x  7  to the graph of
y   x  1 x  3 x  7  ?

12. What transformation could you use to describe the change from the graph of y   x  1 x  2  x  3 x  4  to
the graph of y    x  1 x  2  x  3 x  4  ?

E. Finding Zeros Graphically
Function                                                      Zeros
13. f ( x)  3x  x  4
2

14. f ( x)  x3  3x 2  x  3
15. f ( x)  x3  3x  2
16. f ( x)  x 4  5x3  3x 2  5x  4
17. f ( x)  x 4  x3  7 x 2  x  6
18. f ( x)  x 4  3x3  6 x 2  28x  24
19. f ( x)  x5  2.6 x 4  1.11x3  3.74 x 2  0.73x  0.3

20. What does the Fundamental Theorem of Algebra say about the number of real zeros of a polynomial in
relation to the degree of the polynomial?

21. A fourth degree polynomial has four zeros: Sometimes                    Always    Never.

22. A polynomial can have more zeros than the highest degree of the function. True                   False:

23. What is the greatest number of zeros possible for the function f ( x)  x5  15x3  10 x 2  60 x  72 ?
24. Determine the number of real zeros for f ( x)  x5  15x3  10 x 2  60 x  72 ?

25. What are the real zeros of f ( x)  x5  15x3  10 x 2  60 x  72 ?

F. Rational Root Theorem
If f(x) = anxⁿ +...+ a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form:
p
 , where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.
q

Use the Rational Root Theorem to find the zeros of the polynomial f ( x)  x3  2 x 2  11x  12 . To use your
graphing calculator for this, following the steps outlined below.

Enter this polynomial into the graphing calculator by pressing
! and entering the function

Set the table function to ask you for each x-value by pressing
` ç and matching the screen to the right.

Now, press ` ê and enter each of the ±(p/q) values into the
table. The rational roots will result in y-values of 0.

For each polynomial given, use the Rational Root Theorem to list the possible rational zeros. When you
have the possible zeros, use the handheld to determine which values are actual zeros of the function.

Function                                 Possible Zeros                          Actual Zeros
26. f ( x)  x  4 x  6 x  36 x  27
4      3      2

27. f ( x)  3x3  2 x 2  11x  12
28. f ( x)  10 x 4  3x3  29 x2  5x  8
29. f ( x)  x 4  2 x3  x 2  2 x  2

G. The Remainder Theorem
If a polynomial P  x  of degree n  1 is divided by  x  a  , where a is constant, then the remainder is P  a  .
Use the Remainder Theorem to find P  a  .
30. P( x)  x3  4 x2  8x  6; a  2        31. P( x)  x3  7 x 2  15x  9; a  3          32. P( x)  6 x3  x 2  4 x  3; a  3

Determine whether each binomial is a factor of 3x3  10x2  x 12 .
33. x  3             34. x  4                 35. x  2                               36. x 1                   37. x  5

38. A polynomial P  x  is divided by a binomial x  a . The remainder is zero. What conclusion can you draw?
39. A student represented the product of three linear factors as x3  x2  2x . She used x 1 as one of the factors.
Use division to prove that the student made an error.
40. Suppose that 3, 1 and 4 are zeros of a cubic polynomial function. Sketch a graph. Could there be more than
one graph?

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