Download Polynomial Functions and End Behavior

Unit 5 – Polynomial Functions
DAY
1
2
3
4
5
6
7
8
9
TOPIC
-Vocabulary for Polynomials
-Add/Subtract Polynomials
-Identifying Number of Real Zeros for a graph from
calculator
Multiplying Polynomials
ASSIGNMENT
6.1 # 1-18
Long Division of Polynomials (begin synthetic division)
-Synthetic Division and Synthetic Substitution
-Remainder Theorem
-Synthetic Division and Synthetic Substitution
-Remainder Theorem
REVIEW
QUIZ (50 points)
Factor Theorem
Factoring Higher Degree Polynomials
Sum/Diff of Two Cubes
Grouping
More on 6.4
6.3 # 3, 4, 13, 15, 16
6.3 # 20-22, 24-26, 31,
32, 49
Worksheet
(p.11 in packet)
TO BE ANNOUNCED
13
14
Rational Roots Theorem
Solving Polynomial Equations by Factoring
Multiplicity of Roots
Rational Roots Theorem and Solving Polynomial Equations
with the help of a calculator
-Writing Functions Given Zeros
-Fundamental Theorem of Algebra
-Irrational and Complex Conjugate Roots Theorems
More on 6.6
REVIEW
15
TEST-entire unit
10
11
12
6.2 # 1-8, 10, 18-25
6.4 # 17-23, 34, 35, 50
6.4 26 – 30(skip 27), 33,
34, 36
6.5 # 2-4, 11-13
6.5 # 24-26 (Use RRT),
27-29
6.6 # 1, 2, 15, 16, 18
6.6 # 7, 8, 20, 21
P 474 # 2-54 (even –
this might change)
TBA
Note: Due to Winter Break we will finish up to Day 11 or 12. On
Day 12 we will have a short quiz. We will not have a test on this
unit. Days 11 and 12 will be handled when we return from break.
Page 1 of 28
sU5 Day 1
Polynomial Functions (Section 6.1)
An expression that is a real number, a variable, or a product of a real number and a variable with whole-number
exponents _______________________________
A _______________________ is a monomial or the sum of monomials. Standard form is written in descending
order of exponents.
The exponent of the variable in a term is the ______________________
constant
P( x)  2 x 3  5 x 2  2 x  5
Leading coefficient
cubic term
quadratic term
linear term
Facts about polynomials:
1. classify by the number of terms it contains
2. A polynomial of more than three terms does not usually have a special name
3. Polynomials can also be classified by degree.
4. the degree of a polynomial is: ____________________________________
____________________________________________________________
Degree
Name using
degree
Polynomial
example
Alternate Example
0
-9
11
1
x-4
4x
2
x 2  3x  1
x2  1
3
x3  3x2  10x  7
x3  10x
4
Quartic
5
quintic
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Number of Terms
Monomial/monomial
Trinomial/binomial
Practice
1. Write each polynomial in standard form. Then classify it by degree and by the number of terms.
a. 7 x  5 x 4
b. x 2  4 x  3x3  2 x
d. 3x3  x 2  4 x  2 x3
c. 4x  6x  5
2. ADDING and SUBTRACTING Polynomials. Write your answer in standard form.
a.) ( x  4 x  3x  2 x)  (10 x  4 x  9 x )
2
3
2
3
b.)
(3  2 x2 )  ( x2  6  x)
3. Graph each polynomial function on a calculator.
Read the graph from left to right and describe when it increases or decreases.
Determine the number of x-intercepts. Sketch the graph.
a.) f ( x)  x  x
b.) f ( x)  3x  2 x  1
3
3
XMIN = -5
XMAX = 5
YMIN = -5
YMAX = 5
Description: from left to right the graph
increases, decreases slightly, and increases
again. There are 3 x-intercepts = 3 REAL
ZEROS.
c.) f ( x)  x  8 x  1
4
XMIN = -5
XMAX
Page =3 5
of 28
YMIN = -15
YMAX = 10
2
Description:
d.) f ( x) 
1 4
x  2 x3  2
6
XMIN = -5
XMAX = 5
YMIN = -5
YMAX = 5
Description:
Description:
Closure: Describe in words how to determine the degree of a polynomial.
U5 Day 2
Multiplying Polynomials (Section 6.2)
WARM UP
1-2 Evaluate 3-4 Simplify
1.  2 4
2. (2) 4
5.) x 3  x8
6.) x 2  3x 3
WARM UP Part 2
Multiply
Multiplying Polynomials
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3.) x – 2(3x-1)
4.) 3( y 2  6 y)
Distribute the x and then distribute the 2. Combine like terms and simplify.
Try These
If you are interested in using the Alternate Method (see example below), I set up the first one for you.
a.) (a  3)(2  5a  a )
b.) ( y  7 y  5)( y  y  3)
2
2
-5a
2
2
a2
a
-3
6-2
Alternate Method
– Table
Multiplying
Polynomials
Example 2B: Multiplying Polynomials
Find the product.
(y2 – 7y + 5)(y2 – y – 3)
Multiply each term of one polynomial by each term of
the other. Use a table to organize the products.
y2
–y
–3
The top left corner is the first
y2
y4
–y3 –3y2 term in the product. Combine
terms along diagonals to get
–7y –7y3 7y2 21y
the middle terms. The bottom
right corner is the last term in
5
5y2 –5y –15 the product.
y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15
y4 – 8y3 + 9y2 + 16y – 15
Holt Algebra 2
Page 5 of 28
U5 Day 3
Long Division Polynomials (Section 6.3)
Review Days 1 and 2
Classify the each polynomial by degree and number of terms.
1.  x3  5x2  70
2.  x4  11x3  7 x2  x
Perform the indicated operation.
3. (8x  5x 2 )  ( x 2  6  8x)
4. ( y  5)( y  y  3)
5. 5 xy(10 y  3xy  5 x y)
6. (x – 1) (x – 2) (x + 3)
3
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2
2
Just for fun try the following long division without your calculator
(OH NOOOO!! Please don’t make me think – it’s almost winter break).
3169/15 =
Let’s do one together:
( y 2  2 y3  25) /(y-3)
The Setup:
Write the dividend (the part on the inside) in standard form, including any terms with
a coefficient of 0.
2 y3  y 2  0 y  25
Setup a long division problem the same way you would when dividing numbers.
y – 3 2y3 – y2 + 0y + 25
Practice
1
5. (3x 2  9 x  2) / ( x  )
3
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6.
U5 Day 4 Synthetic Division (Section 6.3 cont.)
Synthetic division is a shorthand method of dividing a polynomial by a linear
binomial by using only the _______________. For synthetic division to work, the
polynomial must be written in standard form, using 0 and a coefficient for any
missing terms, and the divisor must be in the form
(x – a).
In long division we divide and
subtract, in synthetic division we
____________ and ____________.
Let’s Try These Together
Page 8 of 28
Synthetic Substitution – using synthetic
division to evaluate polynomials. Use
the Remainder Theorem.
Example:
P(x) = x3  4x2  3x  5 for x = 4
Try These
U5 Day 5
(Section 6.3 cont.)
Use this time to complete any skipped problems for days 1-4.
Ready to Go On?
Page 9 of 28
6-3 Lesson Practice Quiz
1. Divide by using long division.
( 8x 3  6x 2  7 ) ÷ (x + 2)
2. Divide by using synthetic division.
( x 3  3x  5) ÷ (x + 2)
3. Use synthetic substitution to evaluate
P(x) = ( x 3  3x 2  6) for x = 5 and x = –1.
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4. Find an expression for the height of a
parallelogram whose area is represented by
(2 x 3  x 2  20 x  3) and whose base is
represented by (x + 3).
If time allows start on homework
U5 Day 5 Homework Worksheet – show all work
U5 Day
6 Quiz
Review
Show all work-be organized-write answers on the lines provided.
I. Perform the indicated operation. Write the answer in standard form.
1. (2 x 4  4 x 2  6 x  5 x3  1)  (2 x  9 x 4  8 x3  1x 2  7)  ___________________________________
2. (1x  2 x 4  3x3  4)  (9 x 4  8x 2  4 x  2 x3  3)  ___________________________________
2a) The degree of your answer to #2 is_________ 2b) The leading coefficient in your answer is_______
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Multiply:
3. ( x 2  2 x  4)( x  1)
4. ( x 2  x  1)( x 2  x  1)
______________________________
5. ( x 2  4 x  1)(2 x 2  3)
____________________________
6. Expand ( x  3)3
_______________________________
__________________________________
III. Divide using LONG division: Write the quotient, with the remainder, if there is one, as a fraction, on the
answer line.
7. (3x3  4 x 2  7 x)  ( x  3)
8. ( x3  3x 2  2 x  4)  ( x  1)
____________________________
__________________________
4
2
9. Divide using SYNTHETIC division: ( x  3x  4 x  3)  ( x  2) . Write the quotient, with the remainder, if
there is one, as a fraction.
______________________________
10. If f ( x)  x5  10 x3  3x 2  3x  9 , find f (4) using synthetic division.
f (4)  ______
11. Is ( x  1) a factor of f ( x)  x  5x  2 x  1x  2 ? Explain how you know. Show work.
4
Page 12 of 28
3
2
Fill in the blanks for the chart below.
Example of a function
Degree of the function
4
3
f ( x)  x  5 x  2
Name/type of function
f ( x)  x 5
f ( x)  x 3
f ( x)  7
f ( x)  2 x  3
f ( x)  9 x 2
Complete each statement below.
A polynomial with 2 terms is called a ________________The degree of 3x3  y 2  z 5 is____________.
U5 Day 8 Factoring (Section 6.4)
Warm Up
Factor each expression
a.) 3x – 18y
b.) a 2  b 2
Use the distributive property
a.) (x – 10) (2x + 7)
c.) x3  2 x 2  15 x
b.) (a 2  1)(a  2)
The Remainder Theorem: if a
polynomial is divided by (x – a),
the remainder is the value of the
function at a. So, if (x – a) is a
factor of P(x), then P(a) = 0.
Determine Whether a Linear Binomial is a Factor
Example1: Is (x-3) a factor of P(x) = x 2  2 x  3 .
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Example 2: Is (x + 4) a factor of P(x) =
2 x 4  8 x3  2 x  8
You Try
a.) Is (x+2) a factor of P(x) = 4 x 2  2 x  5 . b.) Is (3x - 6) a factor of P(x) = 3x 4  6 x3  6 x 2  3x  30 .
Note: the binomial is not in the form (x – a)
Factor by Grouping
Common binomial factor – Write as two binomials in simplified form
a.) 2y(5x + 12) + 7(5x + 12)
b. x ( x  3)  4( x  3)
2
c.) 3a (4b  1)  9b(1  4b)
4
Exampe1:
You Try
a.) x  2 x  x  2
3
2
b.)
c.) x3  2 x 2  9 x  18
Graphing Calculator Table Feature (compare
original equation and factored form)
Page 14 of 28
Use the Table feature on your calculator to check problems a and b from above (You Try Section)
3
2
a.) x  2 x  x  2 Which values of Y1 and Y2 are 0?____________
b.)
Which values of Y1 and Y2 are 0?____________
Closure
1. If (x – 3) is a factor of some polynomial P(x) what does that tell you about the remainder?
2. If you divide 5 into 80 what is your remainder? What does this tell you about the number 5 with
regard to the number 20?
U5 Day 9
Factoring continued…
Warm Up
1.) ( x  2)( x 2  2 x  4)
Example 1:
a = ________
2.) (5 y  2)(25 y 2  10 y  4)
8 y3  27
(Identify a and b)
b = ________
Example 3:
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Example 2:
a = ________
b = ________
8  z6
a = ________
b = ________
5x4  40 x
You Try
a.)
2 x4  56 x
b.)
64 x3  1
d.)
250 x4  54 x
e.) Challenge
c.)
3x4  24 x
64 x6  1
Application
Closure
1.) Describe one key difference between factoring the SUM of perfect squares VS the DIFFERENCE
of perfect squares.
Page 16 of 28
U5 Day 10
Real Roots in Polynomial Equations (Section 6.5)
From section 5-3 the Zero _________ Property defines how we can find the roots (or
solutions) of the polynomial equation P(x) = 0 by setting each __________ equal to 0.
Factor
Example 1:
(Factor out the GCF)
Let’s look at the graph.
Use a simple substitution here. I’ll show you.
Example 2:
You Try
a.)
2 x6  10 x5  12 x4  0
b.)
x3  2 x2  25x  50
Multiplicity Calculator Exploration
Multiplicity simply means that a factor is repeated in a polynomial function.
By Definition: The multiplicity of root r is the number of times that x – r is a ___________ of P(x).
1. What is the multiplicity in the following:
M = _____
What does the graph do if M is EVEN?
Compare this to y = ( x  1) M = ______
4
SKETCH THE FUNCTIONS
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y = ( x  1) ?
2
2. . What is the multiplicity in the following:
M = _____
y = ( x  1) ?
3
What does the graph do if M is ODD?
Compare this to y = ( x  1) M = ______
5
SKETCH THE FUNCTIONS
3. What is the multiplicity in the following: y = ( x  1) ( x  4)
3
2
There are two values for M. Let’s see what happens. Do you have a prediction?
SKETCH THE FUNCTION
4. Find the roots and the multiplicity of each root for y = (2x - 10)(x – 7)(x + 1)(x+1)
5. Identify the roots and state the multiplicity for each root: (Use your calculator.)
a.) f(x) = ( x  2)( x  5)( x  1)2
b.)
2 x6  22 x5  48x4  72x3
Closure: How is a real root with odd multiplicity different from a real root with even multiplicity?
Explain (yes in words).
U5 Day 11
Rational Root Theorem (Section 6.5 cont.)
Warm Up
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Example:
Step 3 Test on the possible rational roots. Look at the graph, which one seems possible.
Use Division and the Remainder Theorem to test.
Step 4 List all factors.
Step 5 Find all roots. Set each factor = 0. Sometimes you’ll need the quadratic formula.
(Ignore the numbering.)
Follow the directions. Just practice listing the possible roots.
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Show all work
1. Let f ( x)  2x 4  7 x3  5x 2  28x  12 .
a. List all the possible rational roots. (p/q’s)
b. Use a calculator to help determine which values are the roots and perform synthetic division with
those roots.
c. Write the polynomial in factored form and determine the zeros of the function. List the multiplicity
of each zero. (You will need to use the quadratic formula.)
2. Let g ( x)  x3  5x 2  23x  8 .
a.
List all the possible rational roots. (p/q’s)
b. Use a calculator to help determine which values are the roots and perform synthetic division with
those roots.
c.
Write the polynomial in factored form and determine the zeros of the function. List the multiplicity
of each zero.
Page 20 of 28
U6 Day 12
Fundamental Theorem of Algebra (Section 6.6)
Warm up – Identify real roots
(hints: set = 0; factor or graph on TI)
5
4
1. 4x – 8x – 32x3 = 0
3
2
2. x –x + 9 = 9x
4
2
3. x + 16 = 17x
3
2
4. 3x + 75x = 30x
Example 1: Writing Polynomial Functions
Write the simplest polynomial with roots -2, 2 and 4
Step 1 - Write the binomial factors:
Step 2 – Multiply the binomial factors:
You Try - Write the simplest polynomial with roots:
a) 0, 3, 2/3 (yes that last one is a fraction)
b) -2, 4, 2/3
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Notice that the degree of the function in Example 1 is the same as the
number of _______. This is true for all polynomial functions. However, all of
the zeros are not necessarily ______ zeros. Polynomials functions may
have complex zeros that are not ______ numbers.
Fundamental Theorem of Algebra
Every Polynomial function of degree n  1 has at least ________ __________, where a
zero may be a complex number.
Corollary
Every Polynomial function of degree n  1 has exactly ____ zeros, including
multiplicities.
Example 2: Finding all Roots of a Polynomial Equation
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
The degree of the polynomial:____________________
The number of roots of the polynomial:____________
Use the Rational Root Theorem to find the POSSIBLE ROOTS.
p = _______
q = _______
List ALL possible roots:___________________
There are too many POSSIBLE roots to check. Use the graph.
Looks like x = -1 might be a root. Test it.
How?
Be Careful, if the root is x = -1, what is the
binomial we are dividing
by?___________
How will you know if -1 is a root?
-1| 1 -3 5
-27 -36
Ok, we have one root. Find another.
Let’s complete Example 2 together…
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Be sure to differentiate between factors and roots/solutions.
List the FACTORS for the Example 2 polynomial__________________________
List the ROOTS/SOLUTIONS for the Example 2 polynomial________________
You Try
Solve x 4  4 x3  x 2  16 x  20  0 by finding ALL roots.
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Show work below.
U6 Day 13 Fundamental Theorem of Algebra (Section 6.6 cont…)
This first theorem is actually from 6.5 but we’ll do it here.
Think about this by solving for x in the following: x 2  5  0
Find the conjugate of the following:
a.) 7  2
b.) 11
c.) 1  6
d.)  3
Note: Real Numbers are a subset of the complex numbers (a + bi) where b is
zero so the real number 10 could be represented as 10 + 0i. In the following
theorem the term complex root will only refer to a + bi where b is not 0.
Complex Conjugate Root Theorem
If a + bi is a root of a polynomial equation with real number coefficients then
_________ is also a root.
Write the simplest polynomial function with the given zeros:
a.) 1 + 2i
b.) 2
c.) 1 + 2i,
2
(Return to the previous page
Page 24 of 28
and do 3, 4 and 6.)
Chapter 6 Test Review
1.
Simplify the expression.
a)
3x6 (4 x5  7 x)
2.
Use synthetic division to show that ( x  2) is NOT a factor of (3x3  4 x 2  7 x  4) .
Remember—if the factor is x – 2, use 2 for the synthetic division.
3.
Given that P( x)  2 x3  3x 2  8 , find P (2) .
Solve these polynomial equations by factoring.
4.
x3  2 x 2  8 x  0
5
x3  5 x 2  4 x  20  0
6.
2 x5  12 x 4  18 x3  0
Page 25 of 28
b)
(2 x3  x 2 )( x3  6 x 2 )
7. Write a polynomial equation with the following roots: 2, 4, 1
(answer in standard form—multiplied out)
8. Write a polynomial equation with the following roots: 0, 3i
(leave in factored form)
9. For f ( x)  x 4 ( x  5)( x  3)2 ( x  5)3 , state the roots and their multiplicities. If you multiplied the
function out into standard form (don’t actually do it, unless you’re super bored), what would be the
degree of the function?
10. How many roots (counting multiplicities) does f ( x)  6 x7  10 x 4  3x3  9 x  4 have?
11. There is a 4th degree polynomial that has roots 4  9 5 and 7  i . What are the other 2 roots?
(#12 – 13) Find all the roots of the function. Use your calculator (and the rational root theorem)
to find a root that will work with synthetic division. Once you get the function down to a
quadratic, you can use factoring or the quadratic formula.
12. f ( x)  x3  2 x 2  3x  6
13. f ( x)  x3  19 x  30
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Page 27 of 28
Some answers:
1a) 12 x11  21x 7
1b) 2 x 6  11x5  6 x 4
3. 4
4. 0, 2, -4
5. 2, -2, -5
6. 0, 3
7. x3  3x 2  6 x  8
8. x( x  3i )( x  3i )

x3  9 x
9. 0 w/ mult. of 4, 5 w/ mult of 1, -3
w/ mult of 2,
-5 w/ mult of 3. degree =
10
10. 7
11. 4  9 5, 7  i
12. 2,  3
13. 5, -2, -3
28