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+ Polynomials Chapter 6
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+ Polynomials Chapter 6. + 6.1 - Polynomial Functions.

Jan 17, 2016

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Page 1: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

PolynomialsChapter 6

Page 2: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+6.1 - Polynomial Functions

Page 3: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Objectives

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

Classify polynomials

Model data using polynomial functions

Page 4: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+http://www.youtube.com/watch?v=udS-OcNtSWo

Page 5: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Vocabulary A polynomial is a monomial or the sum of

monomials.

The exponent of the variable in a term determines the degree of that polynomial.

Ordering the terms by descending order by degree. This order demonstrates the standard form of a polynomial. P(x) = 2x³ - 5x² - 2x + 5

Leading Coefficient

Cubic Term

Quadratic Term

Linear Term

Constant Term

Page 6: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ Standard Form of a Polynomial

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

PolynomialStandard Form

Polynomial

Page 7: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Parts of a Polynomial

P(x) = 2x3 – 5x2 – 2x + 5Leading Coefficient:

Cubic Term:

Quadratic Term:

Linear Term:

Constant Term:

Page 8: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Parts of a Polynomial

P(x) = 4x2 + 9x3 + 5 – 3xLeading Coefficient:

Cubic Term:

Quadratic Term:

Linear Term:

Constant Tem:

Page 9: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Classifying Polynomials

We can classify polynomials in two ways:

1) 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

Page 10: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ Classifying Polynomials

2) 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

Page 11: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 12: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ 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

Page 13: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ 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)

Page 14: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Picking a Model

Given Data, we need to decide which type of model is the best fit.

Page 15: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

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

Page 16: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

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

Page 17: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+6.2 - Polynomials & Linear Factors

Page 18: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ 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.

Page 19: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ Factored to Standard

Write the following polynomial in standard form:

(x+1)(x+2)(x+3)

Page 20: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factored to StandardWrite the following polynomial in standard form:

(x+1)(x+1)(x+2)

Page 21: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factored to Standard

Write the following polynomial in standard form:

x(x+5)2

Page 22: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Standard to Factored form

To Factor:

1. Factor out the GCF of all the terms

2. Factor the Quadratic

Example: 2x3 + 10x2 + 12x

Page 23: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Standard to Factored formWrite the following in Factored Form

3x3 – 3x2 – 36x

Page 24: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Standard to Factored form

Write the following in Factored Form

x3 – 36x

Page 25: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+The Graph of a Cubic

Page 26: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 27: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ Relative Max and MinFind the relative max and min of the following polynomials:

1. f(x) = x3 +4x2 – 5x Relative min: Relative max:

2. f(x) = -x3 – 7x2 – 18x Relative min: Relative max:

Calculator:2nd CALC Min or Max

Use a left bound and a right bound for each min or max.

Page 28: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Finding Zeros

When a Polynomial is in factored form, it is easy to find the zeros, or where the graph crosses the x-axis.

EX: Find the Zeros of y = (x+4)(x – 3)

Page 29: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factor Theorem

The Expression x – a is a linear factor of a polynomial if and only if the value a is a zero of the related polynomial function.

Page 30: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Find the Zeros

Find the Zeros of the Polynomial Function.

1. y = (x – 2)(x + 1)(x + 3)

2. y = (x – 7)(x – 5)(x – 3)

Page 31: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Writing a Polynomial Function

Give the zeros -2, 3, and -1, write a polynomial function. Then 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.

Page 32: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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.

Page 33: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Find the Multiplicity of a Zero

Find any multiple zeros and their multiplicity

y = x4 + 6x3 + 8x2

Page 34: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 35: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+6.3 Dividing Polynomials

Page 36: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 37: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Long Division

Divide WITHOUT a calculator!!

1. 2.

Page 38: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Steps for Dividing

Page 39: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using Long Division on Polynomials

Using the same steps, divide.

Page 40: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using Long Division on Polynomials

Using the same steps, divide.

Page 41: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using Long Division on Polynomials

Using the same steps, divide.

Page 42: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Synthetic Division

Page 43: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Synthetic Division

Step 1: Switch the sign of the constant term in the divisor. Write the coefficients of the polynomial in standard form.

Step 2: Bring down the first coefficient.

Step 3: Multiply the first coefficient by the new divisor.

Step 4: Repeat step 3 until remainder is found.

Page 44: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

Use Synthetic division to divide

3x3 – 4x2 + 2x – 1 by x + 1

Page 45: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

Use Synthetic division to divide

X3 + 4x2 + x – 6 by x + 1

Page 46: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

Use Synthetic division to divide

X3 + 3x2 – x – 3 by x – 1

Page 47: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Remainder Theorem

Remainder Theorem: If a polynomial P(x) is divided by (x – a), where a is a constant, then the remainder is P(a).

Page 48: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using the Remainder Theorem

Find P(-4) for P(x) = x4 – 5x2 + 4x + 12.

Page 49: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+6.4 Solving Polynomials by Graphing

Page 50: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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)

Page 51: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

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

Page 52: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solve by Graphing

x3 + 3x2 = x + 3

x3 – 4x2 – 7x = -10

Page 53: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solve by Graphing

x3 + 6x2 + 11x + 6 = 0

Page 54: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solving by Factoring

Page 55: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factoring Sum and Difference

Factoring cubic equations:

Note: The second factor is prime (cannot be factored anymore)

Page 56: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ Factor:

1) x3 - 8

2) 27x3 + 1

Page 57: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+You Try! Factor:

1) x3 + 64

2) 8x3 - 1

3) 8x3 - 27

Page 58: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

Solving a Polynomial Equation

Page 59: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 60: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solve by factoring:

1. 2x3 + 5x2 = 7x

2. x2 – 8x + 7 = 0

Page 61: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using the patterns to Solve

So solve cubic sum and differences use our pattern to factor then solve.

X3 – 8 = 0

Page 62: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using the patterns to Solve

x3 – 64 = 0

Page 63: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using the patterns to Solve

x3 + 27 = 0

Page 64: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

Factoring by Using Quadratic Form

Page 65: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factoring by using Quadratic Formx4 – 2x2 – 8

Page 66: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factoring by using Quadratic Formx4 + 7x2 + 6

Page 67: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Factoring by using Quadratic Formx4 – 3x2 – 10

Page 68: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solving Using Quadratic Form

x4 – x2 = 12

Page 69: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+

6.5 Theorems About Roots

Page 70: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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)

Page 71: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Solutions/Roots

How many solutions will each equation have? What are they?

1. x3 – 6x2 – 16x = 0

2. x3 + 343 = 0

Page 72: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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.

Page 73: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 74: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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.

Page 75: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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).

Page 76: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ 8) x3 – 5x2 + 7x – 35 = 0

Page 77: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+ 10) 4x3 + 16x2 -22x -10 = 0

Page 78: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 79: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Recall

Solve the following by taking the square root:

X2 – 49 = 0

X2 + 36 = 0

Page 80: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Using the Theorems

Given one Root, find the other root!

1. √5 2. -√6

3. 2 – i 4. 2 - √3

Page 81: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 82: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+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

Page 83: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

1. Find a 2nd degree equation with roots ±√7

Page 84: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Examples

1. Find a 2nd degree equation with roots ±2√5

2. Find a 2nd degree equation with roots ±6i

Page 85: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Examples

Find a 2nd degree equation with a root of 7 + i

Page 86: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

Find a 3rd degree equation with roots 4 and 3i

(x - _______)(x - ______)(x - ______)

Page 87: + Polynomials Chapter 6. + 6.1 - Polynomial Functions.

+Example

Find a third degree polynomial equation with roots 3 and 1 + i.