+Warm Up #1

+

PolynomialsUnit 6

+6.1 - Polynomial Functions

+Objectives

By the end of today, you will be able to…

Classify polynomials

Model 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 Coefficient

Cubic Term

Quadratic Term

Linear Term

Constant Term

+ Standard Form of a Polynomial

For example: P(x) = 2x3 – 5x2 – 2x + 5

PolynomialStandard Form

Polynomial

+Parts of a Polynomial

P(x) = 2x3 – 5x2 – 2x + 5

Standard Form:

Leading Coefficient:

Cubic Term:

Quadratic Term:

Linear Term:

Constant Term:

+Parts of a Polynomial

P(x) = 4x2 + 9x3 + 5 – 3xStandard Form:

Leading 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!

Constant

Linear

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 terms

2x3 – 4x2 + 5x + 4

+Classifying Polynomials

Write each polynomial in standard form. Then classify it by degree AND number of terms.

1. -7x2 + 8x5 2. x2 + 4x + 4x3 + 4

3. 4x + 3x + x2 + 5 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)

2. (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.

+

x y0 2.82 54 66 5.58 4

Using a graphing calculator, determine whether a linear, quadratic, or cubic model best fits the values in the table.

Enter the data. Use the LinReg, QuadReg, and CubicReg options of a graphing calculator to find the best-fitting model for each polynomial classification.

Graph each model and compare.

The quadratic model appears to best fit the given values.

Linear model Quadratic model Cubic model

Comparing Models

+

Polynomial

Models

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 Ticket1) Determine which type of model best fits the

values in the table (Linear, Quadratic, or Cubic) and find the model

2) 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:

x -5 -1 0 1 5

y -5 -1 0 1 5

+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 StandardWrite 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 formWrite 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)

X -4 -2 0 2 4

Y 3 1 0 1 3

Graph the points below and decide which model would be best (Linear, Quadratic or Cubic).

Hint – Look at the scatterplot!

+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

2. 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!!

+Steps for Dividing

+Using Long Division on Polynomials

+Divide

+Using Long Division on Polynomials

+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)

+

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

Solving by Graphing: 2nd Way

+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

2) 8x3 - 1

3) 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 Formx4 – 2x2 – 8

+Factoring by using Quadratic Formx4 + 7x2 + 6

+Factoring by using Quadratic Formx4 – 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 are all factors of 24 (the constant term) since a = 1.

+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 2. -√6

3. 2 – i 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

2. 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.