Chapter 2 notes from powerpoints

Similar documents
Algebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions

Skills Practice Skills Practice for Lesson 10.1

Lesson 7.1 Polynomial Degree and Finite Differences

More Polynomial Equations Section 6.4

Warm-Up. Use long division to divide 5 into

Polynomial Functions

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2

Unit 1: Polynomial Functions SuggestedTime:14 hours

Chapter 3: Polynomial and Rational Functions

( 3) ( ) ( ) ( ) ( ) ( )

5.1 Polynomial Functions

Polynomial Functions and Models

Operations w/polynomials 4.0 Class:

Class IX Chapter 2 Polynomials Maths

Chapter Five Notes N P U2C5

a real number, a variable, or a product of a real number and one or more variables with whole number exponents a monomial or the sum of monomials

Chapter 2 Formulas and Definitions:

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

2-2: Evaluate and Graph Polynomial Functions

Characteristics of Polynomials and their Graphs

Using Properties of Exponents

6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomial and Synthetic Division

3 What is the degree of the polynomial function that generates the data shown below?

Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

3 Polynomial and Rational Functions

Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+10).


Lesson 11: The Special Role of Zero in Factoring

S56 (5.1) Polynomials.notebook August 25, 2016

, a 1. , a 2. ,..., a n

Downloaded from

Chapter 2 Polynomial and Rational Functions

Maintaining Mathematical Proficiency

Unit 2 Polynomial Expressions and Functions Note Package. Name:

Math 1310 Section 4.1: Polynomial Functions and Their Graphs. A polynomial function is a function of the form ...

Unit 2 Rational Functionals Exercises MHF 4UI Page 1

Section 3.1: Characteristics of Polynomial Functions

Bell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.

Section 0.2 & 0.3 Worksheet. Types of Functions

Theorems About Roots of Polynomial Equations. Theorem Rational Root Theorem

Polynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1

Math 3 Variable Manipulation Part 3 Polynomials A

Question 1: The graphs of y = p(x) are given in following figure, for some Polynomials p(x). Find the number of zeroes of p(x), in each case.

Higher-Degree Polynomial Functions. Polynomials. Polynomials

b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

Chapter 3-1 Polynomials

Grade 12 Pre-Calculus Mathematics Notebook. Chapter 3. Polynomial Functions

Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Pre-Algebra 2. Unit 9. Polynomials Name Period

Solving Quadratic Equations Review

Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring.

2.1 Quadratic Functions

Procedure for Graphing Polynomial Functions

NAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.

A repeated root is a root that occurs more than once in a polynomial function.

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

Modeling Data. 27 will get new packet. 24 Mixed Practice 3 Binomial Theorem. 23 Fundamental Theorem March 2

Use the Rational Zero Theorem to list all the possible rational zeros of the following polynomials. (1-2) 4 3 2

UNIT 5 VOCABULARY: POLYNOMIALS

Theorems About Roots of Polynomial Equations. Rational Root Theorem

A Library of Functions

MHF4U Unit 2 Polynomial Equation and Inequalities

Midterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k.

A monomial or sum of monomials

Warm Up. Factor each quadratic. 1. x x + 24 = 0

3.3 Real Zeros of Polynomial Functions

Pre-Calculus 12 Note Package

Unit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Power and Polynomial Functions. College Algebra

Chapter 2 Polynomial and Rational Functions

We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

How many solutions are real? How many solutions are imaginary? What are the solutions? (List below):

(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =

Chapter 4E - Combinations of Functions

TEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Chapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

171S4.3 Polynomial Division; The Remainder and Factor Theorems. October 26, Polynomial Division; The Remainder and Factor Theorems

Tropical Polynomials

171S4.3 Polynomial Division; The Remainder and Factor Theorems. March 24, Polynomial Division; The Remainder and Factor Theorems

where a =, and k =. Example 1: Determine if the function is a power function. For those that are not, explain why not.

Polynomial Review Problems

Honors Advanced Algebra Unit 3: Polynomial Functions October 28, 2016 Task 10: Factors, Zeros, and Roots: Oh My!

Factoring Review WS 3.0. Factoring Polynomials Factor and Remainder Theorems. Factoring a Sum or Difference of Cubes Pg. 182 # 1-5

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

Module 11 Lesson 3. Polynomial Functions Quiz. Some questions are doubled up if a pool wants to be set up to randomize the questions.

Honours Advanced Algebra Unit 2: Polynomial Functions Factors, Zeros, and Roots: Oh My! Learning Task (Task 5) Date: Period:

How might we evaluate this? Suppose that, by some good luck, we knew that. x 2 5. x 2 dx 5

1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.

Maths Extension 2 - Polynomials. Polynomials

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Lesson 7.1 Polynomial Degree and Finite Differences

Mid-Chapter Quiz: Lessons 2-1 through 2-3

2 the maximum/minimum value is ( ).

Section 6.6 Evaluating Polynomial Functions

Table of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials

Transcription:

Chapter 2 notes from powerpoints

Synthetic division and basic definitions Sections 1 and 2

Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real numbers. The following function is called a polynomial function of x with degree n. n f ( x) a x a x... a x a x a n n 1 2 n 1 2 1 0 No! No! In polynomial functions, THE EXPONENTS ON THE VARIABLE CANNOT BE FRACTIONS AND CANNOT BE NEGATIVE.

2.1 Polynomials n n 1 2 anx an 1x a2 x a1 x a0 Terms: Coefficients: Leading Term: n a x, a x,..., a x, a x, a Leading Coefficient: Degree: n n n 1 2 n 1 2 1 0 a, a,..., a, a, a n ax n n 1 2 1 0 n a The term containing the highest power of x n The power of x contained ** Note : The polynomial 0 has no degree. in the leading term Constant Term n-1 Term Quadratic Term Linear Term

Classifying Polynomials Polynomials are often classified by their degree. The degree of a polynomial is the highest degree of its terms. Degree Name of Polynomial Function Example Zero f(x) = -3 First f(x) = 2x + 5 Second f(x) = 3x 2 5x + 2 Third f(x) = x 3 2x 1 Fourth Constant Linear Quadratic Cubic Quartic f(x) = x 4 3x 3 +7x-6 Fifth f(x) = 2x 5 + 3x 4 x 3 +x 2 Quintic 5

Any value of x for which P x = 0 is a root of the equation and a zero of the function. State whether the function is a polynomial function. Give the zeros if they exist. P x = x 2 4 f x 2x 3 32x g x 0 = x 2 4 = (x + 2)(x 2) ±2 are the zeros of the function f is a polynomial function..., f x 2x x 16 2x x 4 x 4 2 the zeros are 0, 4 x 1 g is not a polynomial function x 1 the only zero is 1

Main Ideas: Synthetic Division 4 2 Divide 2x 15x 10x 5 by x 3 3 2 0-15 -10 5 6 18 9 3 2 6 3 1 2 4 2 2x 15x 10x 5 3 2 2 2x 6x 3x 1 x 3 x 3

Main Ideas Synthetic Division 4 3 2 Is x 2 a factor? If P x 3x 7x 5x 9x 10, find P 2 4 2 The factor If theorem S x states 3x that 5xif a is 9a xconstant, 10 find, then Sx a2 is a factor of polynomial P x if and only if P a = 0 2 3-7 -5 9 10 P 2 0 6-2-14-10 3-1 -7-5 0 Therefore, x 2 is a factor of P(x)

Main Ideas Synthetic Division 4 2 If S x 3x 5x 9x 10 find, S 2 Is x + 2 a factor? S 2 20 2 3 0-5 9 10-6 12-14 10 3-6 7-5 20 Therefore, x + 2 is not a factor of S(x) The factor theorem states that if a is a constant, then x a is a factor of polynomial P x if and only if P a = 0

Section 2.2 Synthetic Division; The Remainder and Factor Theorems Objective: To use synthetic division and to apply the remainder and factor theorems.

Vocabulary 23 4 5 3 Dividend Divisor Quotient Remainder When 23 is divided by 4, the quotient is 5 and the remainder is 3. 4 is not a factor of 23, because if when 23 is divided by 4 the remainder is not zero.

The Remainder Theorem When a polynomial P(x) is divided by x a, the remainder is P(a) P( x) ( x a) Remainder =P(a)

The Factor Theorem if and only if

SECTion 3- graphing polynomials

Starts Up, Ends Up Starts Down, Ends Down

Starts Down, Ends Up Starts Up, Ends Down

2.3 Graphing Polynomial Functions Describe the general shape of a cubic function. How does the sign of the leading coefficient affect the graph? Graph of f x ax 3 bx 2 cx d

2.3 Graphing Polynomial Functions You can quickly sketch the graph of a cubic function by hand if the function is in factored form. Sketch f x x 1 x 1 x 2. Zeros: -1, 1, 2

2.3 Graphing Polynomial Functions Graph of f x ax 4 bx 3 cx 2 dx e

2.3 Graphing Polynomial Functions Sketch the following graphs. Use a calculator to check your graphs. a) y x x 2 x 1 b) y x 3 x 1 x 1 x 2 c) y x x 2 x 1 d) y x x 3 x 1 x 1 x 2

2.3 Graphing Polynomial Functions

2.3 Graphing Polynomial Functions

2.3 Graphing Polynomial Functions

2.3 Graphing Polynomial Functions

Effect of A Squared Factor 2 P x x c c is a double root of P x 0 The graph of P is tangent to the x-axis at x c. 2 Graph y x 1 x 3 Graph y x 1 x 3 x 4 2 2 Graph y x x 1 x 3 2

Effect of A Squared Factor 2 Graph y x 1 x 3

Homework Section 2.2 Page 61 #1-25 odds

Effect of A Squared Factor Graph y x 1 x 3 x 4 2

Effect of A Squared Factor 2 2 Graph y x x 1 x 3

Effect of A Cubed Factor 3 P x x c c is a triple root of P x 0 P flattens out around x-axis at this point. c,0 and crosses the Graph y x 2 3 3 Graph y x 2 3 Graph y x 1 x 3

Effect of A Cubed Factor 3 P x x c c is a triple root of P x 0 P flattens out around x-axis at this point. Graph y x 2 3 c,0 and crosses the

Effect of A Cubed Factor 3 P x x c c is a triple root of P x 0 P flattens out around x-axis at this point. 3 Graph y x 2 c,0 and crosses the

Effect of A Cubed Factor 3 P x x c c is a triple root of P x 0 P flattens out around x-axis at this point. 3 Graph y x 1 x 3 c,0 and crosses the

Graphing review For test review: 9/26

I can graph polynomials examples Determine end behavior and orientation 1) f x = 6x 7 3x 3 + x 2 12 2) f x = 3x 6 x + 1 Find the x- and y-intercepts 3) y = 4 x 5 2 (x + 1)(2x 5)

Warm Up Determine end behavior and orientation 1) f x = 6x 7 3x 3 + x 2 12 x, f x x, f x 2) f x = 3x 6 x + 1 x, f x x, f x

Warm Up Find the x- and y-intercepts 3) y = 4 x 5 2 (x + 1)(2x 5) X-intercepts y-intercept Use Zero Product Property! x 5 = 0 x + 1 = 0 2x 5 = 0 x = 5 x = 1 x = 5/2 Plug in 0 for x y = 4 0 5 2 0 + 1 2 0 5 y = 4 5 2 (1)( 5) y = 500

y = x 1 2 (x + 1) x 2 1 0 1 2 y

y = (x 1) 2 (x + 1) 2 x 2 1 0 1 2 y

Steps Example: y = 2 x + 1 x 2 (x + 8) 1) Find the roots (x-intercepts) 2) Find the y-intercept x + 1 = 0 x 2 = 0 x + 8 = 0 x = 1 x = 2 x = 8 ( 1,0) (2,0) ( 8,0) y = 2(0 + 1)(0 2)(0 + 8) y = 2 1 2 8 y = 32 (0, 32)

Steps Example: y = 2 x + 1 x 2 (x + 8) 3) Determine orientation and end behavior Odd degree, positive leading term As x, f x As x, f x

Steps 4) Graph 1,0 (2,0) ( 8,0) (0, 32) As x, f x As x, f x Example: y = 2 x + 1 x 2 (x + 8)

Steps 1) Find the roots (x-intercepts) 2) Find the y-intercept Example 2: y = x + 5 (x 1) 2 x + 2 x + 5 = 0 x 1 = 0 x + 2 = 0 x = 5 x = 1 x = 2 (5,0) (1,0) ( 2,0) y = 0 + 5 (0 1) 2 0 + 2 y = 5 ( 1) 2 2 y = 10 (0,10)

Steps 3) Determine orientation and end behavior Example 2: y = x + 5 (x 1) 2 x + 2 Even degree, negative leading term As x, f x As x, f x

Steps 4) Graph 5,0 (1,0) ( 2,0) (0,10) Example: y = x + 5 (x 1) 2 x + 2 As x, f x As x, f x

Steps 1) Find the roots (x-intercepts) 2) Find the y-intercept Example 3: y = x x + 1 (x 3) x = 0 x + 1 = 0 x 3 = 0 x = 0 x = 1 x = 3 (0,0) ( 1,0) (3,0) y = 0 0 + 1 0 3 y = 0 (1) 3 y = 0 (0,0)

Steps Example: y = x x + 1 (x 3) 3) Determine orientation and end behavior Odd degree, negative leading term As x, f x As x, f x

I can graph a polynomial Warm Up Graph: y = x x + 1 (x 3) x intercepts: y intercept: End Behavior:

Steps 1) Find the roots (x-intercepts) 2) Find the y-intercept Example 3: y = x x + 1 (x 3) x = 0 x + 1 = 0 x 3 = 0 x = 0 x = 1 x = 3 (0,0) ( 1,0) (3,0) y = 0 0 + 1 0 3 y = 0 (1) 3 y = 0 (0,0)

Steps Example: y = x x + 1 (x 3) 3) Determine orientation and end behavior Odd degree, negative leading term As x, f x As x, f x

X Intercepts: (0,0) 3,0 ( 1,0) Y Intercept:(0,0) End Behavior: Odd degree, negative leading term As x, f x As x, f x

Objective 2.6 Solving Polynomial Functions by Factoring To solve polynomial equations by various methods of factoring, including the use of the rational root theorem. Grouping Terms 3 2 x x x 5 4 20 0 3 2 x x 4x 5 20 0 2 x x x 5 4 5 0 2 x x 4 5 0 x 2 x 2 x 5 0 x 2, 5

Grouping best to use if: P(x) is a cubic. P(x) has 4 terms. Pair up terms. Factor within the terms Factor again.

Try it: x 3 + 6x 2 4x 24 = 0 x = 6, ±2

x 3 + 6x 2 4x 24 = 0 x 3 + 6x 2 + 4x 24 = 0 x 2 x + 6 4 x + 6 = 0 x + 6 x 2 4 = 0 x + 6 x + 2 x 2 = 0 x + 6 = 0, x 2 = 0, x + 2 = 0 x = 6, x = 2, x = 2

4 2 Solve: 2x x 3 0 by rewriting the quartic in quadratic form. 6, x i 2

2x 4 x 2 3 = 0 Let u = x 2 Then: 2x 4 x 2 3 = 0, becomes: 2u 2 u 3 = 0 2u 3 u + 1 = 0 u = 3, u = 1 2 Remember u = x 2, so solving for x: 3 2 = x2, x = ± 3 2 = 3 2 = 3 2 1 = x 2 so x = ±i 2 = ± 6, and 2 2

When to use quadratic form The polynomial has 3 terms The degree is an even number (though this is not required) The exponent of the middle term is half the degree (necessary) The 3 rd term is constant Examples: x 8 + x 4 + 20, x 22 + x 11 + 15, x 13 + x 6.5 215

Try it: 2x 4 = 7x 2 + 15 x = ± 6 2, ±i 5

Classwork Class Exercises Page 60 #1-5

Homework Page 61 #5,7,13,19,23 Page 66 #3,9,11,13,15,21,23,27,29

Homework Page 83 #1, 3, 9, 10, and 13 34 every 3rd

Classwork Page 83 #1-6

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback