Administrivia. Matrinomials Lectures 1+2 O Outline 1/15/2018
|
|
- Laurence Perkins
- 5 years ago
- Views:
Transcription
1 Administrivia Syllabus and other course information posted on the internet: links on blackboard & at Check assignment sheet for reading assignments and eercises. Be sure to read about polynomials, including roots, factorization, long division and remainders. If you haven t done so already, get and install Freemat and start working on the tutorial. 1 Matrinomials Lectures 1+ O Outline Basics: polynomials and roots Horner s Form and Quick Computation Products and Sums of Roots Reverse Polynomials Sums of Reciprocal Roots Long Division and Remainders Palindromials 1
2 Reminder: What s a polynomial? Polynomials have ROOTS ) 1)(5 ( Roots are related to Factors Complete factorization ).4. )(.4. 1)( ( ) ( i i 6
3 Comple Number Review Etension of the real numbers created by inventing a nonreal number 1. Any combination of the form with a and b real numbers is defined to be a comple number. These form a number system with addition, subtraction, and multiplication defined using the usual rules of algebra. Eg (+7i)+(4-i) = 7 + 4i; ( + 7i)(4 i) = 1 9i + 8i 1i (FOIL) = i + 1 (because i = 1) = + 19i. There is even division. Eg: Historically, there has been a progression of different number systems developed as various types of numbers were recognized and defined: whole numbers, fractions (rational numbers), negative numbers, and irrational numbers (giving us the reals). The comple number system is just another step in this progression. 9 Fundamental Theorem of Algebra Refers to any polynomial of the form where the coefficients,,, are comple numbers and 0. This includes the case that the coefficients are real numbers. The theorem says that such a polynomial can always be epressed in this form: where the roots,,, are comple numbers, not necessarily all distinct. This is an eistence theorem: there must eist n comple roots (again not necessarily distinct). 10 Finding them is another matter.
4 Solutions by Radicals For quadratics, cubics, and quartics all roots epressible from coefficients using addition, subtraction, multiplication, division, and square-, cube-, and fourth-roots. Familiar quadratic formula: Similar formula for roots of 0: Even more complicated formula(s) for quartics. Abel-Galois-Ruffini Theorem: No such formulas eist for quintics and higher degree equations. For eample, it can be shown that the quintic 15 5 has at least one root that is not epressible in terms of radicals. This proves that there is no general formula using radicals for all solutions of all quintics. 1 Wikipedia Ecerpt 14 4
5 Finding Roots is Hard But we can find out some things easily The sum of the roots is 11/5 The average of the roots is 11/0 The sum of the reciprocals of the roots is 7/ We ll get back to roots in a bit First let s look at computation Can you compute p() in your head? 17 Horner s Form Standard descending form Horner form Also referred to as partially factored or nested form 18 5
6 Derivation of Horner Form 1 Quick Evaluation Compute p(): (((5 11) + 6) + 7) Answer = 7 Compute p(): (((5 11) + 6) + 7) Answer = 180 Compute p(/5): (((5 # 11)# + 6)# + 7)# Answer = /15? 6
7 Getting Back to our Roots Coefficients and combinations of roots 5 Product of Roots ±Constant term / highest degree coefficient Eample: (The ± sign is + because degree is even) Product of the roots is /5 6 7
8 For our eample Key Idea of Proof Say the roots are r, s, t, and u. p() = 5( r)( s)( t)( u) Multiply this out to find the constant term 5rstu = - Note constant term is 5(- r)(- s)(- t)(- u), so for odd degree we get an etra sign 9 Sum of Roots nd highest degree coefficient divided by highest degree coefficient Eample: Sum of the roots is Average of the roots is 11/5 11/0 0 8
9 A System To Remember Specified sum and product of two unknowns For eample: this system 5 8. Solution: and y are the roots of the quadratic equation 580. Reason: 58iff 5 and 8. The unknowns in any system of the form must be given by. Eercises 1. Prove the sum of the roots result. Prove this: For any polynomial p, the average of the roots of p is equal to the average of the roots of the derivative p. 4 9
10 Reverse Polynomial Consider the polynomial p( ) 7 11 The reverse polynomial is 6 Rev p( ) Question: How are the roots of Rev p related to the roots of p? Answer: Roots of reverse polynomial are reciprocals of roots of the original. 7 Sum of Reciprocal Roots 1 st degree coefficient divided by the constant coefficient Eample: Sum of the reciprocal roots is 7/ Average of the reciprocal roots is 7/1 8 10
11 Eercises 1. Prove this: For any polynomial p of degree n with nonzero constant term, Rev p() = n p(1/).. Use 1 to prove that the roots of Rev p() are the reciprocals of the roots of p().. Use to prove: if p() = a n n + + a 1 + a 0 and a 0 0, then the sum of the reciprocals of the roots of p is a 1 / a 0 41 Polynomial Long Division Eample: ( 5 + 6) (-) 4 11
12 Polynomial Long Division Redo the eample working from the constants upward 45 Polynomial Long Division Another eample: ( 5 + 6) (-1) (Do it on white board / scratch paper) Answer Similar to the long division 1 to find. Alternate form of answer: -6 ( ) = -6 + /(-1) Part in parentheses is a power series for the rational function 1/(1-). Similar to mied fraction form of answer to a division problem: fraction part = remainder / divisor 46 1
13 Amazing Application Start with p() = + Find the derivative Reverse both (Do it on whiteboard/scratch paper) Do a long division problem of the reversed p() into the reversed p (), working from the constants forward (Do it on whiteboard/scratch paper) (Do it on whiteboard/scratch paper) Answer: The coefficients have an astonishing interpretation: sums of powers of roots 49 Checking the answer p() = + = (-)( 1) Roots are, 1, and -1 Sum of roots = Sum of squares of roots = 6 Sum of cubes = 8 Sum of fourth powers = 18 Etc. 50 1
14 Proof Hints Rev p() = n p(1/) Logarithmic Derivative: f / f = (ln f ) If f () = ( r) ( s) ( t) then (ln f ( ))' r s t Geometric Series: 1 1 1/ a 1 4 and 1 a 1 / a 5 Palindromials p() = reverse p() Recall: if p(r) = 0 (r0) then [rev p](1/r) = 0. So for palindromials, whenever r is a nonzero root, so is 1/r. Eample: and -1 are not roots, so roots come in reciprocal pairs Must factor as (-r)(-1/r)(-s)(-1/s) Rewrite: ( u+ 1) ( v+ 1) where u = r+1/r and v = s + 1/s 54 14
15 Matching Coefficients ( u + 1) ( v + 1) = u + v = -7 and uv + = - Two unknowns. Sum = -7, product = -4 They are the roots of = 0 u and v are given by 7 65 Our factorization is Solve for Use quadratic formula on each factor Roots from first factor are Remaining roots are
16 General Reduction Method p() = a 6 + b 5 + c 4 + d + c + b + a p()= (a + b + c + d + c/ + b/ + a/ ) p()/ =a( +1/ ) + b( +1/ ) + c(+1/)+d We want roots of a( +1/ ) + b( +1/ ) + c(+1/)+d Almost a polynomial in u = (+1/). u = + + 1/ +1/ = u u = + + / + 1/ = + u + 1/ +1/ = u u Leads to a cubic polynomial in u: a(u u) + b(u ) + c(u)+ d Solve for u, and then substitute in u = (+1/) and solve for. Note u + 1 = Eample: 1 Let 1 Substitute, using and If then so 10, giving. Roots obtained from 1and 1similarly. 6 16
17 Eample Make the standard reduction It s another palindromial! Reduce again Solve with quadratic formula Find u: so Solve for u 65 Solve for We have found 4 values for u We know + 1/ = u Solve u + 1 = 0 with quadratic formula for each known u value That gives 8 roots Here is one: 66 17
Section 6.2 Long Division of Polynomials
Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to
More informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationSummer MA Lesson 11 Section 1.5 (part 1)
Summer MA 500 Lesson Section.5 (part ) The general form of a quadratic equation is a + b + c = 0, where a, b, and c are real numbers and a 0. This is a second degree equation. There are four ways to possibly
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do
More informationDay 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
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More informationSection Other Types of Equations
Section.5 - Other Types of Equations The numbers of solutions to a polynomial with n degree, where n is Natural Number, are n solutions. Solving a Polynomial Equation by factoring Eample Solve: = - = (
More informationRational and Radical Expressions and Equations
Rational and Radical Epressions and Equations Secondary Mathematics Page 44 Jordan School District Unit Cluster 7 (AAPR6 and AAPR7): Rational Epressions Cluster 7: Rewrite rational epressions 7 Rewrite
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
More informationReview: 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
Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a
More informationComplex fraction: - a fraction which has rational expressions in the numerator and/or denominator
Comple fraction: - a fraction which has rational epressions in the numerator and/or denominator o 2 2 4 y 2 + y 2 y 2 2 Steps for Simplifying Comple Fractions. simplify the numerator and/or the denominator
More informationAlgebraic Functions, Equations and Inequalities
Algebraic Functions, Equations and Inequalities Assessment statements.1 Odd and even functions (also see Chapter 7)..4 The rational function a c + b and its graph. + d.5 Polynomial functions. The factor
More informationDefine a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:
1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10)
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More informationDay 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions
1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression
More informationa b + c b = a+c a b c d = ac a b c d = a b d a does not exist
Pre-precalculus Boot Camp: Arithmetic with fractions page http://kunklet.peoplcofedu/ Aug, 0 Arithmetic with fractions To add fractions with the same denominator, add the numerators: () a b + c b = a+c
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationLesson 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,
More informationLecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth. If we try to simply substitute x = 1 into the expression, we get
Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth 1. Indeterminate Forms. Eample 1: Consider the it 1 1 1. If we try to simply substitute = 1 into the epression, we get. This is
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More informationMath 119 Main Points of Discussion
Math 119 Main Points of Discussion 1. Solving equations: When you have an equation like y = 3 + 5, you should see a relationship between two variables, and y. The graph of y = 3 + 5 is the picture of this
More informationMore 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
More informationAlgebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background
Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked
More informationAlgebra 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
More informationPolynomial Functions of Higher Degree
SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More information7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
More informationMini Lecture 9.1 Finding Roots
Mini Lecture 9. Finding Roots. Find square roots.. Evaluate models containing square roots.. Use a calculator to find decimal approimations for irrational square roots. 4. Find higher roots. Evaluat. a.
More informationPolynomials and Polynomial Functions
Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationMAC1105-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
More informationTopic: Expressions & Operations AII.1
Topic: Epressions & Operations AII.1 AII.1 The student will identify field properties, aioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets,
More informationPolynomials. 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
More informationEXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections.6 and.) 8. Equivalent Inequalities Definition 8. Two inequalities are equivalent
More informationAlgebra 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
More informationINTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II
Name: Date: INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II Rational functions are simply the ratio of polynomial functions. They take on more interesting properties and have more interesting
More informationSection 3.7: Solving Radical Equations
Objective: Solve equations with radicals and check for extraneous solutions. In this section, we solve equations that have roots in the problem. As you might expect, to clear a root we can raise both sides
More informationAdvanced Algebra 2 - Assignment Sheet Chapter 1
Advanced Algebra - Assignment Sheet Chapter #: Real Numbers & Number Operations (.) p. 7 0: 5- odd, 9-55 odd, 69-8 odd. #: Algebraic Expressions & Models (.) p. 4 7: 5-6, 7-55 odd, 59, 6-67, 69-7 odd,
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationComplete your Parent Function Packet!!!!
PARENT FUNCTIONS Pre-Ap Algebra 2 Complete your Parent Function Packet!!!! There are two slides per Parent Function. The Parent Functions are numbered in the bottom right corner of each slide. The Function
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationCP Algebra 2. Unit 3B: Polynomials. Name: Period:
CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by
More informationChapter 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
More informationLESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More information9.4 Power Series II: Geometric Series
9.4 Power Series II: Geometric Series A particularly important skill to develop for the AP eam, other than checking that you re in RADIAN mode, is to represent certain types of rational functions as a
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More informationChapter 6: Polynomials
Chapter : Polynomials Chapter : Polynomials POLYNOMIALS Definition: A polynomial is an algebraic epression that is a sum of terms, where each term contains only variables with whole number eponents and
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technology c August 2013 Gregg Waterman This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
More informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
More informationMATH 108 REVIEW TOPIC 6 Radicals
Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises
More informationSOLVING QUADRATIC EQUATIONS USING GRAPHING TOOLS
GRADE PRE-CALCULUS UNIT A: QUADRATIC EQUATIONS (ALGEBRA) CLASS NOTES. A definition of Algebra: A branch of mathematics which describes basic arithmetic relations using variables.. Algebra is just a language.
More informationHilbert s theorem 90, Dirichlet s unit theorem and Diophantine equations
Hilbert s theorem 90, Dirichlet s unit theorem and Diophantine equations B. Sury Stat-Math Unit Indian Statistical Institute 8th Mile Mysore Road Bangalore - 560 059 India. sury@isibang.ac.in Introduction
More informationNAME 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
More informationFundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.
.5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root
More informationMath-1010 Lesson 4-2. Add and Subtract Rational Expressions
Math-00 Lesson - Add and Subtract Rational Epressions What are like terms? Like variables: Like powers: y y Multiples of the same variable same base and same eponent. Like radicals: same radicand and same
More informationRational Expressions
CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions 6. Adding and Subtracting Rational Epressions 6.3 Simplifying Comple Fractions 6. Dividing Polynomials:
More informationUnit 5 Evaluation. Multiple-Choice. Evaluation 05 Second Year Algebra 1 (MTHH ) Name I.D. Number
Name I.D. Number Unit Evaluation Evaluation 0 Second Year Algebra (MTHH 039 09) This evaluation will cover the lessons in this unit. It is open book, meaning you can use your textbook, syllabus, and other
More informationSection September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.
Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a
More information8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
More informationAlgebra 1: Hutschenreuter Chapter 11 Note Packet Ratio and Proportion
Algebra 1: Hutschenreuter Chapter 11 Note Packet Name 11.1 Ratio and Proportion Proportion: an equation that states that two ratios are equal a c = b 0, d 0 a is to b as c is to d b d Etremes: a and d
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
More informationMath From Scratch Lesson 37: Roots of Cubic Equations
Math From Scratch Lesson 7: Roots of Cubic Equations W. Blaine Dowler September 1, 201 Contents 1 Defining Cubic Equations 1 2 The Roots of Cubic Equations 1 2.1 Case 1: a 2 = a 1 = 0.........................
More informationChapter 3-1 Polynomials
Chapter 3 notes: Chapter 3-1 Polynomials Obj: SWBAT identify, evaluate, add, and subtract polynomials A monomial is a number, a variable, or a product of numbers and variables with whole number exponents
More informationTable of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v
These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored,
More information5. Determine the discriminant for each and describe the nature of the roots.
4. Quadratic Equations Notes Day 1 1. Solve by factoring: a. 3 16 1 b. 3 c. 8 0 d. 9 18 0. Quadratic Formula: The roots of a quadratic equation of the form A + B + C = 0 with a 0 are given by the following
More informationAnalysis. The student was expected to know and use the Pythagorean theorem to find the missing side. a 2 + b 2 = c 2
Analysis. Correct Answer : meters (m) The student was epected to know and use the Pythagorean theorem to find the missing side. a + b c 8 + 7 64 + 89 89 64 SKILL: Use the Pythagorean theorem to find the
More information5.4 dividing POlynOmIAlS
SECTION 5.4 dividing PolNomiAls 3 9 3 learning ObjeCTIveS In this section, ou will: Use long division to divide polnomials. Use snthetic division to divide polnomials. 5.4 dividing POlnOmIAlS Figure 1
More informationAdding and Subtracting Rational Expressions
Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators
More information3.3 Dividing Polynomials. Copyright Cengage Learning. All rights reserved.
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
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
More informationAlgebra 2 Chapter 3 Part 1 Practice Test 2018
Synthetic divisions in this worksheet were performed using the Algebra App for PCs that is available at www.mathguy.us/pcapps.php. 1) Given the polynomial f x x 5x 2x 24 and factor x 2, factor completely.
More information5.6 Asymptotes; Checking Behavior at Infinity
5.6 Asymptotes; Checking Behavior at Infinity checking behavior at infinity DEFINITION asymptote In this section, the notion of checking behavior at infinity is made precise, by discussing both asymptotes
More informationMath-3. Lesson 3-1 Finding Zeroes of NOT nice 3rd Degree Polynomials
Math- Lesson - Finding Zeroes of NOT nice rd Degree Polynomials f ( ) 4 5 8 Is this one of the nice rd degree polynomials? a) Sum or difference of two cubes: y 8 5 y 7 b) rd degree with no constant term.
More informationTroy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
More informationNotes on Polynomials from Barry Monson, UNB
Notes on Polynomials from Barry Monson, UNB 1. Here are some polynomials and their degrees: polynomial degree note 6x 4 8x 3 +21x 2 +7x 2 4 quartic 2x 3 +0x 2 + 3x + 2 3 cubic 2 2x 3 + 3x + 2 3 the same
More informationx 2e e 3x 1. Find the equation of the line that passes through the two points 3,7 and 5, 2 slope-intercept form. . Write your final answer in
Algebra / Trigonometry Review (Notes for MAT0) NOTE: For more review on any of these topics just navigate to my MAT187 Precalculus page and check in the Help section for the topic(s) you wish to review!
More informationRadical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots
8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More information2.5 Complex Zeros and the Fundamental Theorem of Algebra
210 CHAPTER 2 Polynomial, Power, and Rational Functions What you ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients... and why These topics provide the complete
More informationUnit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More informationINTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE:
INTRODUCTION TO RATIONAL EXPRESSIONS EXAMPLE: You decide to open a small business making gluten-free cakes. Your start-up costs were $, 000. In addition, it costs $ 0 to produce each cake. What is the
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationA BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
More informationBasic methods to solve equations
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:
More informationPre-Algebra 8 Notes Unit 02B: Linear Equations in One Variable Multi-Step Equations
Pre-Algebra 8 Notes Unit 02B: Linear Equations in One Variable Multi-Step Equations Solving Two-Step Equations The general strategy for solving a multi-step equation in one variable is to rewrite the equation
More informationPerforming well in calculus is impossible without a solid algebra foundation. Many calculus
Chapter Algebra Review Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps
More informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
More informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationPolynomial and Rational Functions
Polnomial and Rational Functions 5 Figure 1 35-mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia
More informationFunction Gallery: Some Basic Functions and Their Properties
Function Gallery: Some Basic Functions and Their Properties Linear Equation y = m+b Linear Equation y = -m + b This Eample: y = 3 + 3 This Eample: y = - + 0 Domain (-, ) Domain (-, ) Range (-, ) Range
More information