# Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)

Size: px
Start display at page:

Download "Polynomials. Teachers Teaching with Technology. Scotland T 3. Teachers Teaching with Technology (Scotland)"

Transcription

1 Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T Scotland Polynomials Teachers Teaching with Technology (Scotland)

2 POLYNOMIALS Aim To demonstrate how the TI-8 can be used to facilitate a fuller understanding of polynomials and show clearly the relationship between the algebraic solution and the graphical solution. Objectives Mathematical objectives By the end of this unit you should know: how torecognise features of various polynomials the relationship between the graph of a sitution and the algebra used to describe it the relationship between roots and factors how to factorise and solve polynomial equations how to find approimate roots of a polynomial by decimal search how to find polynomial coefficients Calculator objectives By the end of this unit you should be able to draw graphs using [Y=] alter the display of a graph using [WINDOW] and [ZOOM]. use the [nd][table] function with appropriate setting, using [nd][tblset] : T Scotland Polynomials Page 1 of 9

3 Factorising and Solving Polynomial Equations Calculator skills sheet Using a TI - 8 to assist you in solving polynomials not only reduces the chances of you making a silly error but also makes the whole process much faster. It can also help you gain a fuller understanding of the mathematics. Here is a typical question from a tetbook and a method for solution. Find the roots of = Before going any further check that the [MODE] screen looks like this, particularly ensure that the TI is not rounding answers by highlighting Float. Rounded answers could appear as non-eistent roots. Enter the function on the [Y=] screen and graph the result on the [ZOOM 6:ZStandard] window range. For some functions [ZOOM 4:ZDecimal] is appropriate. The graph of this function shows that it has roots at values which look like -1, and. This can be confirmed by looking at a table of values over this range. Using the [nd][tblset] screen and then [nd][table] we can see that the values of the function are indeed zero at -1, and. Hence we can now say that = -1, = and = are roots. Once we have obtained the roots from the calculator it is easy to say what the factors of the equation are and so this same method can be used to fully factorise a polynomial. The process to be used is almost identical. : T Scotland Polynomials Page of 9

4 Eample 1 Find the roots of = WORKED SOLUTION From graphic calculator root at = No remainder = -1 is a root and ( + 1) is a factor. (Factor Theorem) 5+ 6 This quadratic must factorise to give ( - ) and ( - ). We already know this from the graph and table but can confirm it by multiplying out the bracket or by factorising the quadratic = ( + 1)( ) = = ( + 1)( - )( - ) = + 1 = or - = or - = = - 1 or = or = Eercise 1 Solve these Polynomial Equations = = = = = : T Scotland Polynomials Page of 9

5 Eample Fully factorise This function is a cubic (polynomial of degree ) which means it may have: Roots Factors i.e = a, = b, = c Unique Linear Factors ( - a)( - b)( - c) i.e = a, = b, = b 1 Unique Linear Factor & 1 Repeated pair ( - a)( - b)( - b) 1 i.e = a 1 Linear Factor & 1 Non-factorising Quadratic [ZOOM 6:ZStandard] From the TI it can be seen that this function has 1 unique root and 1 pair of coincidental roots, i.e. it must have 1 unique linear factor and 1 pair of repeated linear factors. The table display shows that the roots are = -1 and = 1, from the graph we can see that the roots at = 1, are coincidental. The roots of this function are = -1, = 1 and = 1. The factors must be ( + 1) and ( - 1) and ( - 1) WORKED SOLUTION From graphic calculator root at = No remainder = -1 is a root and ( + 1) is a factor. (Factor Theorem) This quadratic must factorise to give ( - 1) and ( - 1) = ( + 1)( - + 1) = ( + 1)( - 1)( - 1) : T Scotland Polynomials Page 4 of 9

6 Eample Fully factorise and solve [ZOOM 4:ZDecimal] = From the TI it can be seen that this function has unique roots, i.e. it must have linear factors. From the graph we can see that one of the roots is fractional (about =.5). The table display confirms this showing roots at = -, = -1 and another between = and = 1, since f() = - and f(1) = 8. The roots of this function are = -, = -1 and =.5 The factors must be ( + ) and ( + 1) and ( -.5) WORKED SOLUTION From graphic calculator root at = - By changing the [nd][tbl SET] as shown we can see that the fractional root is indeed at = No remainder = - is a root and ( + ) is a factor. (Factor Theorem) + 1 This quadratic must factorise to give ( + 1) and ( -.5) We do not quote linear factors with fractional values so we can multiply the factor to eliminate the fractional value, in this case by so ( -.5) = ( - 1) Hence the factors of the resulting quadratic are: ( + 1) and ( - 1) = ( + )( - - 1) = ( + )( + 1)( - 1) + = or + 1 = or - 1 = = - or = - 1 or = 1 1 = - or = - 1 or = : T Scotland Polynomials Page 5 of 9

7 Approimate Roots of a Polynomial (Decimal Search) Eample 1 Show that there is a real root of the equation between 1 and, and find an approimation for the root correct to decimal places. + = On your TI enter the function and draw the graph. [ZOOM 4:ZDecimal] From this graph it can be seen that the function does have a root between = 1 and = Using [nd][tbl SET] set up a table of values as shown. The resulting table shows that the root does lies between = 1 and = Using [nd][tbl SET] change the starting value of the table and the step size. The resulting table shows that the root does lies between = 1. and = 1. Change the start and step size again. The resulting table shows that the root does lies between = 1.1 and = 1. Change the start and step size once more. The resulting table shows that the root does lies between = 1.1 and = so to d.p. the root is = 1.1 WORKING f ( ) = + f () 1 = 1 f ( ) = 7 f (.) 1 = 7. f (.) 1 = 497. f (. 11) =. 184 f (. 1) =. 585 f (. 11) =. f (. 114) =. 19 } } } } Root between 1 and, 1 < < Root between 1. and 1., 1. < < 1. Root between 1.1 and 1., 1.1 < < 1. Root is 1. to 1 d.p Root between 1.1 and 1.14, 1.1 < < 1.14 Root is 1.1 to d.p : T Scotland Polynomials Page 6 of 9

8 Finding Polynomial Coefficients Eample 1 If ( + ) is a factor of p , find the value of p. On your TI enter the function using the letter P as the coefficient of on the [Y=] screen. Note to obtain the letter P use [ALPHA][P] The TI will evaluate the function for P= the value stored as P in the memory of the calculator. Store the value zero as P. [][STO][ALPHA][P][ENTER] Using [nd][tblset] and [nd][table] obtain a value of this function when = -, the root which would accompany the factor ( + ). The value of the function when P = and = - is -7. Therefore the term P must equal 7, to make the value at = - be If P = 7, when = - then P = This answer can now be checked by storing the value as P. [][STO][ALPHA][P][ENTER] The table of results now shows that = - is a root and therefore ( + ) is a factor. WORKING Let so f ( ) = P g()= f ( ) = g() + P ( ) 4 4 f ( ) has a factor + f ( ) has a root at = f ( ) = g( ) + P( ) = but g( ) = 7 f ( ) = 7+ 9P = 9P = 7 P = : T Scotland Polynomials Page 7 of 9

9 Eercise (Like Eercise 1 only trickier) Solve these Polynomial Equations = = = 19+ = = Eercise = Eercise Fully factorise these polynomials Show that = has a root between and 1. Find the value of this root to 1 d.p.. Show that = has a root between and 1, and another between -1 and. Find the values of these roots to 1 d.p.. Show that = has a root between 1.5 and. Find the value of this root to d.p. 4. Find all of the roots of = to 1 d.p. 5. Find all of the roots of = to 1 d.p. 6. Find all of the roots of = to 1 d.p. Eercise 5 1. Given that + is a factor of - + k + 6 = Find the value of k.. Given that + 1 is a factor of a = Find the value of a.. Given that + 4 is a factor of p = Find the value of p and solve the equation. 4. Given that + - is a factor of p + q = Find the value of p and q. Hence factorise this equation fully. (HINT: Try to form a pair of simultaneous equations) : T Scotland Polynomials Page 8 of 9

### For each learner you will need: mini-whiteboard. For each small group of learners you will need: Card set A Factors; Card set B True/false.

Level A11 of challenge: D A11 Mathematical goals Starting points Materials required Time needed Factorising cubics To enable learners to: associate x-intercepts with finding values of x such that f (x)

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### Higher. Polynomials and Quadratics 64

hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

### Integrating algebraic fractions

Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Roots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)

Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### Section 3-3 Approximating Real Zeros of Polynomials

- Approimating Real Zeros of Polynomials 9 Section - Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros

### March 29, 2011. 171S4.4 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

### 3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### is identically equal to x 2 +3x +2

Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### The Factor Theorem and a corollary of the Fundamental Theorem of Algebra

Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### Systems of Equations Involving Circles and Lines

Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

### MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Definition 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 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Unit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials

Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial

### To add fractions we rewrite the fractions with a common denominator then add the numerators. = +

Partial Fractions Adding fractions To add fractions we rewrite the fractions with a common denominator then add the numerators. Example Find the sum of 3 x 5 The common denominator of 3 and x 5 is 3 x

### Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### MEP Y8 Practice Book A. In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below: ( ) = +

8 Algebra: Brackets 8.1 Epansion of Single Brackets In this section we consider how to epand (multiply out) brackets to give two or more terms, as shown below: = + 3 + 6 3 18 First we revise negative numbers

### Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 )

Quiz: Factoring by Graphing Question 1a of 14 ( 2 Identifying the roots of a polynomial and their importance 91008 ) (x-3)(x-6), (x-6)(x-3), (1x-3)(1x-6), (1x-6)(1x-3), (x-3)*(x-6), (x-6)*(x-3), (1x- 3)*(1x-6),

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### SAMPLE. Polynomial functions

Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

### UNCORRECTED PAGE PROOFS

number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

### Mathematics as Problem Solving The students will demonstrate the ability to gather information from a graphical representation of an equation.

Title: Another Way of Factoring Brief Overview: Students will find factors for quadratic equations with a leading coefficient of one. The students will then graph these equations using a graphing calculator

### Unit 3: Day 2: Factoring Polynomial Expressions

Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored

### Assessment Schedule 2013

NCEA Level Mathematics (9161) 013 page 1 of 5 Assessment Schedule 013 Mathematics with Statistics: Apply algebraic methods in solving problems (9161) Evidence Statement ONE Expected Coverage Merit Excellence

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division

### Section 5.0A Factoring Part 1

Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### MESSAGE TO TEACHERS: NOTE TO EDUCATORS:

MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please find suggested lesson plans for term 1 of MATHEMATICS Grade 11 Please note that these lesson plans are to be used only as a guide and teachers

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### 2.5 Zeros of a Polynomial Functions

.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

### Slope & y-intercept Discovery Activity

TI-83 Graphing Calculator Activity Slope & y-intercept Discovery Activity Justin Vallone 11/2/05 In this activity, you will use your TI-83 graphing calculator to graph equations of lines. Follow the steps

### Zeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.

_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial

### Sect. 1.3: Factoring

Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion

Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

### 2.4 Real Zeros of Polynomial Functions

SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower

### CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

### 3.6 The Real Zeros of a Polynomial Function

SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### ROUTH S STABILITY CRITERION

ECE 680 Modern Automatic Control Routh s Stability Criterion June 13, 2007 1 ROUTH S STABILITY CRITERION Consider a closed-loop transfer function H(s) = b 0s m + b 1 s m 1 + + b m 1 s + b m a 0 s n + s

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### 63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

### 3.2 The Factor Theorem and The Remainder Theorem

3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### is identically equal to x 2 +3x +2

Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### 1. Which of the 12 parent functions we know from chapter 1 are power functions? List their equations and names.

Pre Calculus Worksheet. 1. Which of the 1 parent functions we know from chapter 1 are power functions? List their equations and names.. Analyze each power function using the terminology from lesson 1-.

### Paper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel

Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table

.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price

Math 1314 Lesson 8 Business Applications: Break Even Analysis, Equilibrium Quantity/Price Three functions of importance in business are cost functions, revenue functions and profit functions. Cost functions

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

### Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

### Answer Key Building Polynomial Functions

Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,

### 03 The full syllabus. 03 The full syllabus continued. For more information visit www.cimaglobal.com PAPER C03 FUNDAMENTALS OF BUSINESS MATHEMATICS

0 The full syllabus 0 The full syllabus continued PAPER C0 FUNDAMENTALS OF BUSINESS MATHEMATICS Syllabus overview This paper primarily deals with the tools and techniques to understand the mathematics

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

Factoring Quadratic Trinomials Student Probe Factor x x 3 10. Answer: x 5 x Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials. Part 1 of the lesson consists

### Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

### FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### FACTORING ax 2 bx c WITH a 1

296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In

### MATH 110: College Algebra

MATH 110: College Algebra Introduction Required Materials Course Components Final Exam Grading Academic Policies Study Suggestions Course Outline and Checklist Introduction Welcome to Math 110. This course