+

PolynomialsChapter 6

+6.1 - Polynomial Functions

+Objectives

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

Classify polynomials

Model data using polynomial functions

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

+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

+ 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 + 5Leading Coefficient:

Cubic Term:

Quadratic Term:

Linear Term:

Constant Term:

+Parts of a Polynomial

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

Cubic Term:

Quadratic Term:

Linear Term:

Constant Tem:

+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

+ 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

+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

+ 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

+

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

+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 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.

+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)

+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.

+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)

+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.

+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

+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!!

1. 2.

+Steps for Dividing

+Using Long Division on Polynomials

Using the same steps, divide.

+Using Long Division on Polynomials

Using the same steps, divide.

+Using Long Division on Polynomials

Using the same steps, divide.

+Synthetic Division

+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.

+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

+Example

Use Synthetic division to divide

X3 + 3x2 – x – 3 by x – 1

+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).

+Using the Remainder Theorem

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

+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.