+ Warm Up #1
+
Polynomials
Unit 6
+ 6.1 - Polynomial
Functions
+
Objectives
By the end of today, you will be able to…
 Classify
 Model
polynomials
data using polynomial functions
+ Vocabulary
A
polynomial is a monomial or the sum of monomials.
 The
highest exponent of the variable determines the
degree of that polynomial.
 standard
form of a polynomial - Ordering the terms by
degree in descending order
P(x) = 2x³ - 5x² - 2x + 5
Leading Cubic
Coefficient Term
Quadratic
Term
Linear
Term
Constant
Term
+ Standard Form of a Polynomial
For
example: P(x) = 2x3 – 5x2 – 2x + 5
Polynomial
4x - 6x + 5
3x 3 + x 2 - 4x + 2x 3
6 - 2x 5
x 3 - 2x 2 - 3x 4
Standard Form
Polynomial
+
Parts of a Polynomial
P(x) = 2x3 – 5x2 – 2x + 5
Standard
Leading
Form:
Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Term:
+
Parts of a Polynomial
P(x) = 4x2 + 9x3 + 5 – 3x
Standard
Leading
Form:
Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Term:
+ Classifying Polynomials
1) By the degree of the polynomial (or the
largest degree of any term of the
polynomial.
Degree
Name
Example
0
Constant
7
1
Linear
2x + 5
2
Quadratic
2x2
3
Cubic
2x3 – 4x2 + 5x + 4
4
Quartic
x4 + 3x2
5
Quintic
3x5 – 3x + 7
+ Graphs are based on degrees!
Linear
Constant
Quadratic
Cubic
Quartic
+
Classifying Polynomials
We can classify polynomials in two ways:
2) By the number of terms
# of Terms
Name
Example
1
Monomial
3x
2
Binomials
2x2 + 5
3
Trinomial
2x3 + 3x + 4
4
Polynomial with 4 2x3 – 4x2 + 5x + 4
terms
+
Classifying Polynomials
Write each polynomial in standard form. Then
classify it by degree AND number of terms.
1.
-7x2 + 8x5
3. 4x + 3x + x2 + 5
2. x2 + 4x + 4x3 + 4
4. 5 – 3x
+Review – Regression Models
1)
Find a linear model for the data below (STAT  CALC  LinReg
2)
Find a quadratic model for the data
(STAT  CALC  QuadReg)
+ Cubic Regression
We have already discussed regression for linear
functions, and quadratic functions. We can also
determine the Cubic model for a given set of points
using Cubic Regression.

STAT  Edit

x-values in L1, y-values in L2

STAT CALC

6:CubicReg
+ Cubic Regression
Find the cubic model for each function:
1.
(-1,3), (0,0), (1,-1), (2,0)
1.
(10, 0), (11,121), (12, 288), (13,507)
+
Picking a Model
Given Data, we need to decide which
type of model is the best fit.
+
Comparing Models
Using a graphing calculator, determine whether a linear, quadratic, or
cubic model best fits the values in the table.
x
0
2
4
6
8
y
2.8
5
6
5.5
4
Enter the data. Use the LinReg, QuadReg, and CubicReg
options of a graphing calculator to find the best-fitting model
for each polynomial classification.
Linear model
Graph each model and compare.
Quadratic model
Cubic model
The quadratic model appears to best fit the given values.
+
Models
Polynomial
You have already used lines and parabolas to
model data. Sometimes you can fit data more
closely by using a polynomial model of degree
three or greater.
Using a graphing calculator, determine
whether a linear model, a quadratic
model, or a cubic model best fits the
values in the table.
x
0
5
10
15
20
y
10.1
2.8
8.1
16.0
17.8
+ Exit Ticket
1)
1)
Determine which type of model best fits the values in
the table (Linear, Quadratic, or Cubic) and find the
model
x
-5
-1
0
1
5
y
-5
-1
0
1
5
Write 2x(3x2 + 4x +1) in standard form. Then classify it
by degree and number of terms.
1) Standard Form:
2) Degree:
3) Classify by degree:
4) Number of Terms:
5) Classify by number of terms:
+ Coming up…
HW tonight – Worksheet 6.1
Unit 6 TEST – Wednesday, April 16th
(possibly Thursday 4/17)
Be prepared for a quiz at any time!!
+ Warm Up # 2
+ HW Check – 6.1
2) y = .013x3 - .174x2 + .795x + 3.125; when x = 7, y = 4.64
3) 5x + 2 ; Linear binomial
6) 5s4 – 2s + 1 ; Quartic trinomial
9) 2x2 – 1 ; Quadratic binomial
12) 3x3
15) a5 + a4 + a3 ; Quintic trinomial
18) 9c4 ; Quartic monomial
21) s2 + 2/3 ; Quadratic binomial
24) 3x + 5
25) y = .26x2 – 3.62x + 29.3 ; average benefit if 2005 is $955.82
26) y = .13x + 2.06 ; 12 days
+
6.2 - Polynomials & Linear
Factors
+ Factored Form
 The
Factored form of a polynomial is a
polynomial broken down into all linear factors.
 We
can use the distributive property to go
from factor form to standard form.
+ Factored to Standard
Write the following polynomial in standard form:
(x+1)(x+2)(x+3)
+ Factored to Standard
Write the following polynomial in standard form:
(x+1)(x+1)(x+2)
+ Factored to Standard
Write the following polynomial in standard form:
x(x+5)2
+
Standard to Factored form
To Factor:
1.
Factor out the GCF of all the terms
2.
Factor the Quadratic
Example: 2x3 + 10x2 + 12x
+ Standard to Factored form
Write the following in Factored Form
3x3 – 3x2 – 36x
+
Standard to Factored form
Write the following in Factored Form
x3 – 36x
+
The Graph of a Cubic
+
•
Vocabulary
Relative Maximum: The greatest Y-value of the
points in a region.
 Relative
Minimum: The least Y-value of the points in a
region.
 Zeros: Place
where the graph crosses x-axis
 y-intercept:
Place where the graph crosses y-axis
+ Relative Max and Min
f(x) = x3 +4x2 – 5x

Relative min:

Relative max:
Calculator:
2nd  CALC  Min or Max
Use a left bound and a right
bound for each min or max.
+ Finding Zeros – from a graph
Locate the x-intercepts
+
Warm Up (Do on the back of your
warm up sheet)
Graph the points below and decide which model
would be best (Linear, Quadratic or Cubic).
Hint – Look at the scatterplot!
X
-4
-2
0
2
4
Y
3
1
0
1
3
+
QUIZ Time! 20 minutes maximum!
+
To find zeros (x-intercepts)
– Set each factor = 0 and solve for x.
Find the Zeros of the Polynomial Function.
1.
y = (x – 2)(x + 1)(x + 3)
2. y = (x – 7)(x – 5)(x – 3)
+ Writing a Polynomial Function
Give the zeros -2, 3, and -1, write a
polynomial function in factored form.
Then rewrite it in standard form to classify it
by degree and number of terms.
+ Give the zeros 5, -1, and -2,
write a polynomial function.
Then classify it by degree and
number of terms.
+
Repeated Zeros
A repeated zero is called a MULITIPLE ZERO.
A multiple zero has a MULTIPLICITY equal to
the number of times the zero repeats.
+
Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
y = x4 + 6x3 + 8x2
+ Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
1.
y = (x – 2)(x + 1)(x + 1)2
1.
y = x3 – 4x2 + 4x
+ Warm Up #3
+ Homework Check – 6.2
+ 6.3 Dividing Polynomials
+ Vocabulary

Dividend: number being divided

Divisor: number you are dividing by

Quotient: number you get when you divide
Remainder: the number left over if it does not
divide evenly

Factors: the DIVISOR and QUOTIENT are
FACTORS if there is no remainder

+
Long Division
Divide WITHOUT a calculator!!
3 4935
+
Steps for Dividing
4 78495
+ Using Long Division on Polynomials
x - 3 x + 3x -12
2
+
Divide
2x -1 2x + 3x - 4x + x +1
4
3
2
+ Using Long Division on Polynomials
x + 4 x + 6x + 8
2
+
Synthetic Division
+
Synthetic Definition
To divide by a linear factor, you
can use a simplified process
that is known as synthetic
division. In synthetic division,
you omit all variables and
exponents.
+
Synthetic Division Steps:
1.
Switch the sign of the constant term in the
divisor. Write the coefficients of the
polynomial in standard form.
2.
Bring down the first coefficient.
3.
Multiply the first coefficient by the new
divisor.
4.
Repeat step 3 until remainder is found.
+
Example
Use Synthetic division to divide
3x3 – 4x2 + 2x – 1 by x + 1
+
Example
Use Synthetic division to divide
X3 + 4x2 + x – 6 by x + 1
+
Check your work!
Dividend = Divisor x Quotient + Remainder
+
Example
Use Synthetic division to divide
X4 + 4x2 + x – 6 by x + 1
+
Example
Use Synthetic division to divide
X3 + 3x2 – x – 3 by x – 1
+
Remainder Theorem
If a polynomial P(x) is divided by
(x – a), where a is a constant, then
the remainder is P(a).
+ Find the remainder for
P(x) = x4 – 5x2 + 4x + 12 divided by (x + 4)
using the Remainder Theorem
+
6.4 Solving Polynomials by
Graphing
+ Solving by Graphing: 1st Way
Solutions are zeros on a graph
Step 1: Solve for zero on one side of the
equation.


Step 2: Graph the equation

Step 3: Find the Zeros using 2nd  CALC
(Find each zero individually)
+
Solving by Graphing: 2nd Way
 Step
1: Graph both sides of the equal sign
as two separate equations in y1 and y2.
 Use
2nd  CALC  Intersect to find the x
values at the points of intersection
+ Solve by Graphing
x3 + 3x2 = x + 3
x3 – 4x2 – 7x = -10
+ Solve by Graphing
x3 + 6x2 + 11x + 6 = 0
+ Solving by Factoring
+ Factoring Sum and Difference
Factoring cubic equations:
Note: The second factor is prime (cannot be
factored anymore)
+ Factor:
1)
x3 - 8
2)
27x3 + 1
+ You Try! Factor:
1)
x3 + 64
1)
8x3 - 1
1)
8x3 - 27
+
Solving a Polynomial Equation
+ Solving By Factoring
Remember: Once a polynomial is in factored
form, we can set each factor equal to zero and
solve.
4x3 – 8x2 + 4x = 0
+ Solve by factoring:
1. 2x3 + 5x2 = 7x
2. x2 – 8x + 7 = 0
+
Using the patterns to Solve
So solve cubic sum and differences use our
pattern to factor then solve.
X3 – 8 = 0
+ Using the patterns to Solve
x3 – 64 = 0
+ Using the patterns to Solve
x3 + 27 = 0
+
Factoring by Using Quadratic Form
+
Factoring by using Quadratic Form
x4 – 2x2 – 8
+
Factoring by using Quadratic Form
x4 + 7x2 + 6
+
Factoring by using Quadratic Form
x4 – 3x2 – 10
+ Solving Using Quadratic Form
x4 – x2 = 12
+
6.5 Theorems About Roots
+
The Degree
Remember: the degree of a polynomial is the
highest exponent.
The Degree also tells us the number of Solutions
(Including Real AND Imaginary)
+ Solutions/Roots
How many solutions will each equation
have? What are they?
1.
x3 – 6x2 – 16x = 0
2.
x3 + 343 = 0
+ Solving by Graphing
Solving by Graphing ONLY works for
REAL SOLUTIONS. You cannot find
Imaginary solutions from a Graph.
Roots: This is another word for zeros or
solutions.
+
Rational Root Theorem
If p/q is a rational root (solution) then:

p must be a factor of the constant
and

q must be a factor of the leading
coefficient
+ Example
x3 – 5x2 - 2x + 24 = 0
Lets look at the graph to find the solutions
 Factored
 (x + 2)(x – 3)(x – 4) = 0
 Note: Roots
since a = 1.
are all factors of 24 (the constant term)
+ Example
24x3 – 22x2 - 5x + 6 = 0
 Lets
look at the graph to find the solutions:
 Factored
 (x + ½ )(x – ⅔)(x – ¾ ) = 0

1,2, and 3 (the numerators) are all factors of 6 (the constant).

2, 3, and 4 (the denominators) are all factors of 24 (the leading
coefficient).
+
8) x3 – 5x2 + 7x – 35 = 0
+ 10) 4x3 + 16x2 -22x -10 = 0
+
Irrational Root Theorem
Square Root Solutions come in PAIRS:
If x2 = c then x = ± √c
If √ is a solution so is -√
Imaginary Root Theorem
If a + bi is a solution, so is a – bi
+
Recall
Solve the following by taking the square
root:
X2 – 49 = 0
X2 + 36 = 0
+
Using the Theorems
Given one Root, find the other root!
1.
√5
3. 2 – i
2. -√6
4. 2 - √3
+ Zeros to Factors
If a is a zero, then (x – a) is a factor!!
When you have factors
(x – a)(x – b) = x2 + (a+b)x + (ab)
SUM
PRODUCT
+ Examples
1.
Find a 2nd degree equation with roots 2 and 3
(x - _______)(x - ______)
2. Find a 2nd degree equation with roots -1 and 6
+ Example
1.
Find a 2nd degree equation with roots ±√7
+
Examples
1.
Find a 2nd degree equation with roots ±2√5
1.
Find a 2nd degree equation with roots ±6i
+
Examples
Find a 2nd degree equation with a root of 7 + i
+ Example
Find a 3rd degree equation with roots 4 and 3i
(x - _______)(x - ______)(x - ______)
+ Example
Find a third degree polynomial equation
with roots 3 and 1 + i.