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

Size: px
Start display at page:

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

Transcription

1 Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the left of the division sign.) 3x 3 17x 2 +15x 25 1) x 5 a) I need to change the sign of the (-5) to positive for my synthetic division Answer: 3x3 17x 2 +15x 25 = 3x 2 2x + 5 remainder 0 x 5 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 3x 3 17x x 25 = (x-5)(3x 2 2x + 5) I the 3x 2 2x + 5 is prime, so this is completely factored. Answer: 3x 3 17x x 25 = (x-5)(3x 2 2x + 5) 3) 4x 3 +8x 2 9x 18 x+2 3a) I need to change the sign of the 2 to negative for my synthetic division Answer: 4x3 +8x 2 9x 18 = 4x 2 9 remainder 0 x+2 3b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 4x 3 + 8x 2 9x 18= (x+2)(4x 2 9) I just need to factor more Answer: 4x 3 + 8x 2 9x 18= (x+2)(2x+3)(2x-3)

2 5) 3x 3 16x 2 72 x 6 5a) I need to change the sign of the (-6) to positive for my synthetic division. I need to think of the numerator having the form 3x 3 16x 2 + 0x Answer: 3x3 16x 2 72 = 3x 2 + 2x + 12 remainder 0 x 6 5b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true 3x 3 16x 2 72 = (x-6)(3x 2 + 2x + 12) The 3x 2 + 2x + 12 is prime, so I can t factor more. Answer: 3x 3 16x 2 72 = (x-6)(3x 2 + 2x + 12) 7) (5x 3 + 6x + 8) (x + 2) this is the same as 5x3 +6x+8 x+2 a) I need to change the sign of the 2 to negative for my synthetic division. I need to insert a 0x 2 term in the numerator 5x 3 + 0x 2 + 6x Answer: (5x 3 + 6x + 8) (x + 2) = 5x 2 10x + 26 remainder -44 7b) skip this part since the remainder is not 0.

3 9) (x 3 27) (x 3) this is the same as x3 27 x 3 a) I need to change the sign of the (-3) to positive for my synthetic division I need to insert a 0x 2 and a 0x. (x 3 + 0x 2 + 0x -27) (x-3) Answer: (x 3 27) (x 3) = x 2 + 3x + 9 remainder of 0 9b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true x 3 27 = (x-3)(x 2 + 3x + 9) The x 2 + 3x + 9 is prime Answer: x 3 27 = (x-3)(x 2 + 3x + 9)

4 #11 20: a) use your graphing calculator, or the rational root theorem to find a zero of the polynomial i) you need to find one zero for a third degree polynomial ii) you need to find two zeros for a fourth degree polynomial b) use synthetic division to completely factor the polynomial (use double synthetic division for fourth degree polynomials) c) Use your answer to part b to solve f(x) = 0 11) f(x) = x 3 + 2x 2 5x 6 here is a graph of f(x) 11a) Answer: I will use the numbers (-1) for my synthetic division, I could have also used 2. 11b) since x = -1 is a zero, I know (x+1) is a factor of f(x) the synthetic division will get me the remaining factors The result of my synthetic division gives me x 3 +2x 2 5x 6 x+1 = x 2 + x 6 remainder 0 so now I can factor f(x) f(x) = x 3 + 2x 2 5x 6 = (x+1)(x 2 +x-6) Answer #11b: f(x) = (x+1)(x-2)(x+3)

5 11c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+1)(x-2)(x+3) = 0 x + 1 = 0 x 2 = 0 x + 3 = 0 x = -1 x = 2 x = -3 Answer #11c: x = -1, 2, -3

6 13) f(x) = 2x 3 13x x 9 here is a graph of f(x) 13a) I will use 3 is the value for my synthetic division 13b) since x = 3 is a zero, I know (x-3) is a factor of f(x) the results of my synthetic division should help me get additional factors of f(x) The result of the synthetic division tells me 2x3 13x 2 +24x 9 x 2 Now I can factor f(x) = 2x 3 13x x 9 = (x-3)(2x 2 7x + 3) Answer 13b: f(x) = (x-3)(x-3)(2x-1) or (x-3) 2 (2x-1) = 2x 2 7x + 3 remainder 0

7 13c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x-3)(x-3)(2x-1) = 0 x - 3 = 0 x 3 = 0 2x 1 = 0 x = 3 x = 3 2x = 1 x = ½ Answer 13c: x = 3, ½

8 15) f(x) =x 3 5x 2 4x - 20 here is a graph of f(x) 15a) I will use (-2), but I could have used any of the three x-intercepts. 15b) since x = (-2) is a zero I know (x+2) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of my synthetic division tells me f(x) = x 3 5x 2 4x - 20 = (x+2)(x 2-7x+10) (I just need to factor the second parenthesis to get my answer) Answer #15b: f(x) = (x+2)(x-2)(x-5) 15c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+2)(x-2)(x-5) = 0 x + 2 = 0 x 2 = 0 x 5 = 0 x = -2 x = 2 x = 5 Answer #15c: x = -2,2,5

9 17) f(x) = 2x x x 2 9x 45 here is a graph of f(x) 17a) I will use (-5) and (1) and perform double synthetic division 17b) since x = (-5) is a zero I know (x+5) is a factor of f(x) since x = 1 is a zero I know (x-1) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of the double synthetic division tells me f(x) = 2x x x 2 9x 45 = (x+5)(x-1)(2x 2 + 9x + 9) Answer: f(x) = (x+5)(x-1)(2x+3)(x+3)

10 17c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+5)(x-1)(2x+3)(x+3) = 0 x + 5 = 0 x 1 = 0 2x + 3 = 0 x + 3 = 0 x = -5 x = 1 2x = -3 x = -3 x = -3/2 Answer #17c: x = -5,1, 3 2,-3

11 19) f(x) = x 4 + 7x 2 8 here is a graph of f(x) 19a) I will use (-1) and (1) and perform double synthetic division since x = (-1) is a zero I know (x+1) is a factor of f(x) since x = 1 is a zero I know (x-1) is a factor of f(x) the results of my synthetic division should help me get more factors of f(x) The result of the double synthetic division tells me Answer: f(x) = x 4 + 7x 2 8 = (x+1)(x-1)(x 2 + 8) this is the answer as the x is prime

12 19c) Solve f(x) = 0 Just take answer to part b and set it equal to 0 and solve for x. (x+1)(x-1)(x 2 + 8) = 0 x+1 = 0 x 1 = 0 x = 0 x = -1 x = 1 x 2 = -8 x = ± 8 = ±2i 2 Answer #19c: x = 1, 1, ±2 2i

13

### More Polynomial Equations Section 6.4

MATH 11009: More Polynomial Equations Section 6.4 Dividend: The number or expression you are dividing into. Divisor: The number or expression you are dividing by. Synthetic division: Synthetic division

### Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers

### Polynomial and Synthetic Division

Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1

### Advanced Math Quiz Review Name: Dec Use Synthetic Division to divide the first polynomial by the second polynomial.

Advanced Math Quiz 3.1-3.2 Review Name: Dec. 2014 Use Synthetic Division to divide the first polynomial by the second polynomial. 1. 5x 3 + 6x 2 8 x + 1, x 5 1. Quotient: 2. x 5 10x 3 + 5 x 1, x + 4 2.

### Skills Practice Skills Practice for Lesson 10.1

Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).

### Cumulative Review. Name. 13) 2x = -4 13) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Cumulative Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Evaluate the algebraic expression for the given value or values of the variable(s).

### 3.4. ZEROS OF POLYNOMIAL FUNCTIONS

3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find

2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.3 Real Zeros of Polynomial Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Use long

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

Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10

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

MA109, Activity 34: Review (Sections 3.6+3.7+4.1+4.2+4.3) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that you can solve the types of problems listed in Activities 33 and

### Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.

Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph

### Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions P(x) = a nx n + a n-1x n-1 + + a 1x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

### 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.

Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope

### SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION

2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides

### Algebra Summer Review Packet

Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills

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

2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using

### Chapter 2 notes from powerpoints

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

### x 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.

1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the

### PreCalculus: Semester 1 Final Exam Review

Name: Class: Date: ID: A PreCalculus: Semester 1 Final Exam Review Short Answer 1. Determine whether the relation represents a function. If it is a function, state the domain and range. 9. Find the domain

### Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

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

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental

### Ch. 12 Higher Degree Equations Rational Root

Ch. 12 Higher Degree Equations Rational Root Sec 1. Synthetic Substitution ~ Division of Polynomials This first section was covered in the chapter on polynomial operations. I m reprinting it here because

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8

### October 28, S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Mission 1 Simplify and Multiply Rational Expressions

Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

TEXTBOOK HELP Pg. 313 Chapter 3.2-3.4 REVIEW ANSWER KEY 1. What qualifies a function as a polynomial? Powers = non-negative integers Polynomial functions of degree 2 or higher have graphs that are smooth

### 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.

Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. -1, 2 5 pts possible Ch 2A Big Ideas 1 Questions

### MEMORIAL UNIVERSITY OF NEWFOUNDLAND

MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS Section 5. Math 090 Fall 009 SOLUTIONS. a) Using long division of polynomials, we have x + x x x + ) x 4 4x + x + 0x x 4 6x

### Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

### Extra Polynomial & Rational Practice!

Extra Polynomial & Rational Practice! EPRP- p1 1. Graph these polynomial functions. Label all intercepts and describe the end behavior. 3 a. P(x = x x 1x. b. P(x = x x x.. Use polynomial long division.

### 2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero

Unit 2 Examples(K) 1. Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

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

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

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

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Solving Polynomial and Rational Inequalities Algebraically. Approximating Solutions to Inequalities Graphically

10 Inequalities Concepts: Equivalent Inequalities Solving Polynomial and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.6) 10.1 Equivalent Inequalities

### Just DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :

5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull

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

hapter 04 Test Name: ate: 1 For the polynomial function, describe the end behavior of its graph. The leading term is down. The leading term is and down.. Since n is 1 and a is positive, the end behavior

### 3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial

3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine

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

2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution

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

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

### Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x ) A) x = 5 B) x = -6 C) x = -5 D) x = 6

Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x - 2) + 6 = 2x - 14 1) A) x = 5 B) x = -6 C) x = -5 D) x = 6 Solve the linear equation.

### 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:

R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)

3.3 Dividing Polynomials Copyright Cengage Learning. All rights reserved. Objectives Long Division of Polynomials Synthetic Division The Remainder and Factor Theorems 2 Dividing Polynomials In this section

### 171S4.4 Theorems about Zeros of Polynomial Functions. March 27, 2012

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Section Properties of Rational Expressions

88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

### Learning Objectives. Zeroes. The Real Zeros of a Polynomial Function

The Real Zeros of a Polynomial Function 1 Learning Objectives 1. Use the Remainder and Factor Theorems 2. Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function 3.

### 3.3 Real Zeros of Polynomial Functions

71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2

### 1. Division by a Monomial

330 Chapter 5 Polynomials Section 5.3 Concepts 1. Division by a Monomial 2. Long Division 3. Synthetic Division Division of Polynomials 1. Division by a Monomial Division of polynomials is presented in

### ALLEN PARK HIGH SCHOOL F i r s t S e m e s t e r A s s e s s m e n t R e v i e w

Algebra First Semester Assessment Review Winter 0 ALLEN PARK HIGH SCHOOL F i r s t S e m e s t e r A s s e s s m e n t R e v i e w A lgeb ra Page Winter 0 Mr. Brown Algebra First Semester Assessment Review

### Introduction. A rational function is a quotient of polynomial functions. It can be written in the form

RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,

### Dividing Polynomials: Remainder and Factor Theorems

Dividing Polynomials: Remainder and Factor Theorems When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is zero, then the divisor is a factor of the dividend.

### Characteristics of Polynomials and their Graphs

Odd Degree Even Unit 5 Higher Order Polynomials Name: Polynomial Vocabulary: Polynomial Characteristics of Polynomials and their Graphs of the polynomial - highest power, determines the total number of

### Name: Class: Date: ID: A

Name: Class: Date: ID: A Algebra Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine which binomial is not a factor of 4x 4 1x 3 46x + 19x

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

Graph and analyze each function. Describe its domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 2x 3 Evaluate the function for several

### Power and Polynomial Functions. College Algebra

Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.

### Section 4.1: Polynomial Functions and Models

Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial

### Concept Category 4. Polynomial Functions

Concept Category 4 Polynomial Functions (CC1) A Piecewise Equation 2 ( x 4) x 2 f ( x) ( x 3) 2 x 1 The graph for the piecewise Polynomial Graph (preview) Still the same transformations CC4 Learning Targets

### H-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.

H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify

### Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,

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

Honors Advanced Algebra Name Unit 3: Polynomial Functions October 8, 016 Task 10: Factors, Zeros, and Roots: Oh My! MGSE9 1.A.APR. Know and apply the Remainder Theorem: For a polynomial p(x) and a number

Zeros and Roots of a Polynomial Function Return to Table of Contents 182 Real Zeros of Polynomial Functions For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent:

### 6.4 Division of Polynomials. (Long Division and Synthetic Division)

6.4 Division of Polynomials (Long Division and Synthetic Division) When we combine fractions that have a common denominator, we just add or subtract the numerators and then keep the common denominator

### PreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College

PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions

### Chapter 8. Exploring Polynomial Functions. Jennifer Huss

Chapter 8 Exploring Polynomial Functions Jennifer Huss 8-1 Polynomial Functions The degree of a polynomial is determined by the greatest exponent when there is only one variable (x) in the polynomial Polynomial

### CHAPTER 4: Polynomial and Rational Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Factors, Zeros, and Roots

Factors, Zeros, and Roots Solving polynomials that have a degree greater than those solved in previous courses is going to require the use of skills that were developed when we previously solved quadratics.

### Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

### 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.

Name: Class: Date: ID: A Midterm Review Short Answer 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. a) b) c) 2. Determine the domain and range of each function.

### Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review

Name: Class: Date: Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review 1 Write 3x 2 ( 2x 2 5x 3 ) in standard form State whether the function is even, odd, or neither Show your work

### 3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

### Review Topics. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review Topics MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. On the real number line, label the points with the given coordinates. 1) 11,- 11 1)

### Math 110 Midterm 1 Study Guide October 14, 2013

Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R

### 2 the maximum/minimum value is ( ).

Math 60 Ch3 practice Test The graph of f(x) = 3(x 5) + 3 is with its vertex at ( maximum/minimum value is ( ). ) and the The graph of a quadratic function f(x) = x + x 1 is with its vertex at ( the maximum/minimum

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

Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor

### Practice Test - Chapter 2

Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

### Chapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions

Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:

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

Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1

### Chapter 3 Polynomial Functions

Trig / Coll. Alg. Name: Chapter 3 Polynomial Functions 3.1 Quadratic Functions (not on this test) For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph. 1.

### Lesson 2.1: Quadratic Functions

Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard

### Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4)

CHAPTER 5 QUIZ Tuesday, April 1, 008 Answers 5 4 1. P(x) = x + x + 10x + 14x 5 a. The degree of polynomial P is 5 and P must have 5 zeros (roots). b. The y-intercept of the graph of P is (0, 5). The number

### Using Properties of Exponents

6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY

### Test # 3 Review. È 3. Compare the graph of n 1 ÎÍ. Name: Class: Date: Short Answer. 1. Find the standard form of the quadratic function shown below:

Name: Class: Date: ID: A Test # 3 Review Short Answer 1. Find the standard form of the quadratic function shown below: 2. Compare the graph of m ( x) 9( x 7) 2 5 with m ( x) x 2. È 3. Compare the graph

### Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations

Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following

### Rational number = one real number divided by another, as in 2/3, = constant/constant

RATIONAL FN S & EQN S CHEAT SHEET MATH1314/GWJ/08 Rational = ratio = fraction Rational number = one real number divided by another, as in 2/3, = constant/constant Generalize constants to polynomials: p(x)/q(x)

### Polynomial Functions and Models

1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

### SB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM.

Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1 Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (-1,4) and

### 2-4 Zeros of Polynomial Functions

Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### Chapter 1- Polynomial Functions

Chapter 1- Polynomial Functions WORKBOOK MHF4U W1 1.1 Power Functions MHF4U Jensen 1) Identify which of the following are polynomial functions: a) p x = cos x b) h x = 7x c) f x = 2x, d) y = 3x / 2x 0

### Learning Target: I can sketch the graphs of rational functions without a calculator. a. Determine the equation(s) of the asymptotes.

Learning Target: I can sketch the graphs of rational functions without a calculator Consider the graph of y= f(x), where f(x) = 3x 3 (x+2) 2 a. Determine the equation(s) of the asymptotes. b. Find the

### 3.4 The Fundamental Theorem of Algebra

333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial

### MAC1105-College Algebra

MAC1105-College Algebra Chapter -Polynomial Division & Rational Functions. Polynomial Division;The Remainder and Factor Theorems I. Long Division of Polynomials A. For f ( ) 6 19 16, a zero of f ( ) occurs

### MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,

### Math 46 Final Exam Review Packet

Math 46 Final Exam Review Packet Question 1. Perform the indicated operation. Simplify if possible. 7 x x 2 2x + 3 2 x Question 2. The sum of a number and its square is 72. Find the number. Question 3.

### Practice Test - Chapter 2

Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

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

1 Algebra II Chapter 5 Test Review Standards/Goals: F.IF.7.c: I can identify the degree of a polynomial function. F.1.a./A.APR.1.: I can evaluate and simplify polynomial expressions and equations. F.1.b./

### Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x

### Synthetic Division. Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z.

Synthetic Division By: Vicky Chen, Manjot Rai, Patricia Seun, Sherri Zhen S.Z. What is Synthetic Division? Synthetic Division is a simpler way to divide a polynomial by a linear factor. You can consider