Name:______________
Block:______________
Algebra2 Unit 5
Polynomials
Unit 5: Polynomial Functions
Day 1: More factoring patterns,
Day 2: Operations & Composition of polynomials
Day 3: Evaluating & Graphing polynomial functions
Day 4: Fundamental Theorem of Algebra
Day 5 & 6: Review and Quiz
Day 7: Applying the Remainder and Factor Theorems
Day 8: Finding all rational zeros
Day 9 & 10: Review and Test
Day 1: Factoring Sum & Difference of Cubes
In these notes we will
Review factoring & extend our knowledge to sums and differences of cubes
So we can
Combine any polynomials and find their zeros.
FACTORING GUIDE:
1. Look for a GCF
- If we have one, factor it out but KEEP IT as a factor.
2. Is what remains a BINOMIAL?
- Is it a difference of TWO SQUARES?
- Is it the SUM OR DIFFERENCE OF TWO CUBES?
3. Is what remains a TRINOMIAL?
- Factor as an easy trinomial if quadratic coefficient = 1.
- Factor by grouping otherwise.
FACTOR the following COMPLETELY.
1. 2x3 – 8x
4. 5x2 + 15x + 10
2. 4x2 – 64
3. 4x2 – 81
5. 10x2 – 25x – 15
ANOTHER PATTERN: SUM and DIFFERENCE of PERFECT CUBES
Factoring Sums or Differences of Two CUBES
a3 + b3 = (a + b)(a2 – ab + b2)
[SOAP the signs….
Same, Opposite, Always Positive]
a3 – b3 = (a – b)(a2 + ab + b2)
Let's verify one algebraically working backwards….
(a – b)(a2 + ab + b2) =
Let’s list are CUBES that you are expected to learn:
Factor the following COMPLETELY using the pattern above.
1)
x3 + 8
2)
x3 – 1
3)
125x3 - 64
4)
x3 – 4
5)
343x3 + 216
6)
64x3 - 27
7)
y3 – 64x3
8)
2x3 + 250
Day 2: Operations & Composition of Polynomials
In these notes we will
Define operations on polynomials: +, -, X quadratics, cubics and more!
So we can
Combine any polynomials and find their zeros.
A polynomial function is a function of the form: f ( x)  a n xn  a n  1 xn  1 
 a1 x  a 0
Note: all coefficients must be real numbers and allexponents of variables must be whole numbers!!
an is the ______________________, n is the ___________, and a0 is the _______________
A polynomial function is in STANDARD FORM if its terms are written in descending order of exponents
from left to right.
Example: f(x) = 2x3 – 5x2 – 4x + 7 leading coefficient_____, constant_____, and degree_____.
Determine if the given functions are polynomial functions. Ask yourself the following questions.
Are the Exponents whole numbers? ____
Are the Coefficients real numbers? ____
If so, write it in standard form and state its degree and leading coefficient.
Function
Polynomial:
YES or NO
Standard form of the
polynomial
y = – 8 -3x5 – 6x2 + 3x
1
y  x 2 1
f ( x)  2 x
g(x) = - 4x2
y = 1 - 5x
y  3x 1
3. Write a polynomial function of degree 7 with leading coefficient of 2
Degree of
polynomial
Leading
Coefficient of
the polynomial
Operations with Polynomials – Adding, Subtracting, Multiplying
1. (2x2 - 8) + (x3 + 4x2 - 12x + 4)
2. (2x2 - 8) – (x3 + 4x2 - 12x + 4)
3. (2z2 – 1)(z + 5)
4. (3x2 + 7)2
5. (x – 3)(x2 + 3x + 9)
6. (a – 6)(a + 5)(a + 1)
Operations with Polynomials – Composition of Functions
**Composing functions is a fancy way of embedding one function into another. When you consider
the DOMAIN of a composition function, f(g(x)), you have to look at restrictions for the input
function AND the result.
1. Let f ( x)  x  7
a. Evaluate f  g (1)  .
and
Let g ( x)  2 x  3
Evaluate g  f (1)  .
Now let’s create a WHOLE NEW function…This is just another way to substitute.
b. Find f  g ( x)  using f ( x)  x  7 and g ( x)  2 x  3 .
Domain: (Look at the input and the result.)
Find f  g (1)  using your new function above.
Is this result the same as in part (a)?
c. Find g  f ( x)  using f ( x)  x  7 and g ( x)  2 x  3 .
Domain: (Look at the input and the result.)
Find g  f (1)  using your new function above.
Is this result the same as in part (a)?
2. Let f ( x)  x 2  2
and
Let g ( x)  x  4 .
Evaluate f  g (5)  .
Evaluate g  f (5)  .
Find f g  f  g ( x)  .
Find g f  g  f ( x)  .
Domain:
Domain:
Check f  g (5)  .
Check g  f (5)  .
Some pairs of functions have a special relationship with respect to composition…..
x 1
y
3
3. Let f(x) = 3x - 1
Let g(x) =
Find f o g = f(g(x))
Find g o f = g(f(x))
Domain:
Domain:
We will learn more about these functions in another unit…but these are called inverse functions.
Day 3: Evaluating & Graphing POLYNOMIAL FUNCTIONS
In these notes we will
Learn a NEW method of evaluating a polynomial and examine polynomial end-behavior
So we can
Sketch and understand polynomial graphs.
y
You know how to graph this….
f(x) = x2 – 2x – 8
x
What is the fn value at x= 0?
How about x= -1?
Evaluating Polynomial Functions – why do we want to do this?
Evaluate by Direct Substitution (Method 1)
Use Direct Substitution to
evaluate f(x) = -3x3 + x2 – 12x – 5 when x = -2
Find f(-2)
In other words,
Evaluate by Synthetic Substitution (Method 2)
Use Synthetic Substitution to evaluate f(x) = -3x3 + x2 – 12x – 5 when x = -2 (same problem as above)
In other words,
Find f(-2)
(Note this polynomial is in STANDARD FORM!)
Multiply
Add
SO……f(
-2
)=
-3
1
-12
-5
1.
2.
3.
4.
Bring Down
Multiply and Add
Multiply and Add
Multiply and Add…
Use Synthetic Substitution to evaluate f(x) = -2x4 – x3 + 4x – 5 when x = -1
In other words,
**Write the coefficients of f(x) in order of descending exponents (there is 0x2)
Multiply
Add
SO……f(
-1
-2
-1
0
4
-5
)=
Use Synthetic Substitution to evaluate f(x) = -2x4 + 3x3 - 8x + 13 when x =2
In other words,
SO……f(
)=
Use Synthetic Substitution to evaluate f(x) = x3 - 216 when x = 3
In other words,
SO……f(
)=
SO, synthetic substitution can also be…….
IF we now have a quick way to FIND AND GRAPH POINTS of a polynomial and TEST FOR ZEROS,
how do we GRAPH THE REST?
Zeros and Multiplicity
Example:
Polynomial function in FACTORED form
f(x) = (x + 4)2 (x + 1) (x – 3)
Degree: ___________ LC: _________
Zeros and multiplicity:
1.

When a factor (x – k) is raised to an ODD power,

When a factor (x – k) is raised to an EVEN power,
the graph CROSSES the x-axis at x = k.
the graph BOUNCES on the x-axis at x = k.
y = 2 (x – 1)2 (x + 2)
Graph Sketch
 y
Degree: ________

Leading Coefficient: ________
End behavior:
x  , f(x)  _____
x  , f(x)  _____
_______
x





Zeros and multiplicity:
_______

_______


Double root(s): _____________________


Analyzing Graphs Graphing with a focus on End Behaviors
Sketch the general shape of each function. Use your calculator!
f(x) = 2x
f(x) = x + 3
f(x) = x3
f(x) = 2x3 + 1
Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______
f(x) = −2x
f(x) = −x + 3
f(x) = −x3
f(x) = −2x3 + 1
Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______
f(x) = 2x2
f(x) = x2+ 2x + 3
f(x) = x4
f(x) = x4+2x3- 1
Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______
f(x) = −2x2
f(x) = −x2 + 3
f(x) = −x4
f(x) = −x4+3x3- x+ 1
Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______
What do you notice with the graphs:
An odd degree function with a positive leading
coefficient:
An odd degree function with a negative
leading coefficient:
An even degree function with a positive
leading coefficient:
An even degree function with a negative
leading coefficient:
We have to be more technical with these descriptions.
Describe the degree (even or odd) and leading coefficient (positive or negative) of the polynomial
function. Then describe the end behavior of the graph of the polynomial function.
1.
2.
Degree _______
Leading Coefficient __________
Degree _______
Leading Coefficient __________
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
3. y = -3x5 – 6x2 + 3x – 8
4. h(x) = 6x8 – 7x5 + 4x
Degree _______
Leading Coefficient __________
Degree _______
Leading Coefficient __________
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
Day 4: Fundamental Theorem of Algebra
In these notes we will
Introduce the FUNDAMENTAL THEOREM OF ALGEBRA and review turning points
So we can "READ" a polynomial – know how many times it crosses the x-axis, and understand its shape
REVIEW:
Solve the following by using the quadratic formula.
2. x2 – 2x – 1 = 0
x2 + 4 = 0
1.
REMEMBER: Because of the ± there will always be two solutions from the formula.
The Fundamental Theorem of Algebra
An nth degree polynomial function has exactly ____ solutions.
1. How many solutions does the equation x4 + 8x2 – 2x + 2 = 0 have? ______
2. How many zeros does the function g(x) = x3 – x2 – 3x – 3 have? _____
Complex and Irrational Conjugates – Imaginary Roots (i) and Radicals (
) always come in
pairs
Complex Conjugate Theorem: if a  bi is a zero, then a  bi is also a zero
Irrational Conjugates Theorem: if a  b is a zero, then a  b is also a zero
Examples:
Given a zero, identify another zero
1.  5
_________
2. 6  2i
________
3. 3  2 7 _______
4. 3i ___________
Given Factors, Identify the Zeros
1. (x + 5)(3x – 2) _________
2. (x + 6)2 _________
3. 3x(x – 2) _________
Given a List of Zeros, Write as a List of Factors
1. 7 __________
2. x = 5, -8, -2 ______________________
3. x =  6 ______________________
4. x =
5. x = 
7
__________________
3
2
__________________
3
6. x = 
1
, 5 _____________________
4
Write a Polynomial Function given the Zeros
Find the standard form of the polynomial with a leading coefficient of 1 that has the given zeros: 4, 1, -2
Write f(x) in factored form:
Change to Standard Form:
Find the standard form of the polynomial with a leading coefficient of 1 that has the given zeros: 5, 3,  2i
Identify all zeros: _________________________
Write f(x) in factored form:
Change to Standard Form:
Graphs of polynomials
Graph f(x) = x3 + x2 - 3x - 6 using your calculator.
Sketch its graph below.
a. Determine the number of zeros for the polynomial _________________
# of real zeros:__________
# of imaginary zeros:____________
b. Determine the number of turning points _________________________
c.
Identify all minimum points. _________________________________________________
d. Identify all maximum points. _________________________________________________
e. Over what intervals is f(x) Decreasing___________________________________________
f.
Over what intervals is f(x) Increasing___________________________________________
g. Describe the end behavior of the graph:
As x   , f ( x)  _____
as x   , f ( x)  _____
Graph f(x) = 3x4 + x3 - 10x2 + 2x + 7 using your calculator.
Sketch its graph below.
a. Determine the number of zeros for the polynomial _________________
# of real zeros:__________
# of imaginary zeros:____________
b. Determine the number of turning points _________________________
c.
Identify all minimum points. _________________________________________________
d. Identify all maximum points. _________________________________________________
e. Over what intervals is f(x) Decreasing___________________________________________
f.
Over what intervals is f(x) Increasing___________________________________________
g. Describe the end behavior of the graph:
As x   , f ( x)  _____
as x   , f ( x)  _____
Day 7: Applying the Remainder and Factor Theorems
In these notes we will
Learn about how to find zeros using polynomial division.
So we can sketch the graph of a polynomial.
Dividing Polynomials:
There are 2 ways to divide polynomials: long division and synthetic division.
Synthetic Division:
2
Divide x 3  2 x  6 x  9 by x – 2
1. First, the divisor must be in the form: (x – k)
is k in the divisor (x - 2)?___
What
2. Write down the coefficients of each term. If a term is
missing, you must use a zero as a place holder.
3. Then drop, multiply and add, multiply and add, etc.
4. The answer is found by filling in the terms
backwards. The last term is the remainder.
Let’s verify by looking at long division.
You try: Divide f(x) = x4 + 4x3 + 16x – 35 by x + 5 using SYNTHETIC division.
Remainder and Factor Theorem
Factor a polynomial
Example:
Factor f(x) = x3 – 2x2 – 40x – 64 completely given that x – 8 is a factor.
STEPS:
1. Synthetic Division - because x – 8 is a factor, f(8) = 0, and x = 8 is a solution.
2. Write as a product of 2 factors.
3. Then completely factor.
Factor: f(x) = x3 – 2x2 – 40x – 64 = (
)(
)(
)
Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial,
f(x) = x3 – 2x2 – 40x – 64
Example:
Factor f(x) = 3x3 – 4x2 – 28x – 16 completely given that x + 2 is a factor.
Factor: f(x) = 3x3 – 4x2 – 28x – 16 =
Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial,
f(x) = 3x3 – 4x2 – 28x – 16.
Find the Zeros, Roots, or Solutions
EXAMPLE:
One zero of f(x) = x3 + x2 – 16x – 16 is 4. Find all the zeros.
1. Synthetic Division – because f(4) = 0, x = 4 is a zero/solution, and x - 4 is a factor
2. Write as a product of 2 factors. Then find the zeros.
Remember you know multiple methods to find the zeros/solutions of a quadratic function.
Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial,
f(x) = x3 + x2 – 16x – 16
EXAMPLE:
One zero of f(x) = x3 + x2 + 2x + 24 is -3. Find all the zeros.
Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial,
f(x) = x3 + x2 + 2x + 24
CHALLENGE:
One zero of f(x) = x4 + 3x2 - 4 is 1. Find all the zeros.
Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial,
f(x) = x4 + 3x2 - 4.
In these notes we will
Day 8: Finding all rational zeros
Learn about how to find ALL zeros without being given a zero or factor to start the problem.
So we can sketch the graph of a polynomial.
What if no zeros are given to start? How can you find all zeros?
Next year you will learn the Rational Zero Theorem that will give you possible zeros in case you do not
have access to the graphing calculator.
LUCKILY…you will use your calculator to graph the polynomial function. Using the graph you will identify a
zero to “test” with synthetic division.
Find All Real Zeros:
Example:
Find all zeros of f(x) = x3 + 2x2 - 11x – 12
1. Graph the polynomial function in the calculator and estimate the real zeros.
2. Test one of your zeros using synthetic division. Remember…it is a solution if the remainder = 0.
3. Write as a new polynomial. Then find the remaining zeros (factor, quadratic formula, or complete the square).
Example:
Find all zeros of f(x) = x3 - 8x2 + 11x + 20
1. Sketch the graph and estimate the real zeros.
2. Test one of your zeros using synthetic division. Remember…it is a solution if the remainder = 0.
3. Write as a new polynomial. Then find the remaining zeros (factor, quadratic formula, or complete the square).
Example:
Find all zeros of f(x) = x3 + 3x2 - 4 (follow all steps used above)
Example:
Example:
Find all zeros of f(x) = x3 + x2 - 3x - 6
Find all zeros of f(x) = 4x3 - 12x2 - x + 15