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


 Rudolf McBride
 2 years ago
 Views:
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 TI8 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 noneistent 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 Nonfactorising 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: miniwhiteboard. 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 xintercepts with finding values of x such that f (x)
More informationQuadratic Functions. Teachers Teaching with Technology. Scotland T 3. Symmetry of Graphs. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T 3 Scotland Quadratic Functions Symmetry of Graphs Teachers Teaching with Technology (Scotland) QUADRATIC FUNCTION Aim To
More information3.3. 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
More informationJUST 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
More informationAn Insight into Division Algorithm, Remainder and Factor Theorem
An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the
More informationPolynomial 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
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationCHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONS
CHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONS EXERCISE 143 Page 369 1. Solve the simultaneous equations graphically: y = 3x 2 y = x + 6 Since both equations represent straightline graphs, only two coordinates
More informationSTRAND: 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
More informationHigher. 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
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : NLYTICL METHODS FOR ENGINEERS Unit code: /0/0 QCF Level: Credit value: OUTCOME TUTORIL LGEBRIC METHODS Unit content Be able to analyse and model engineering situations and solve problems using algebraic
More informationIntegrating 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
More informationPOLYNOMIAL 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
More informationActivity 6. Inequalities, They Are Not Just Linear Anymore! Objectives. Introduction. Problem. Exploration
Objectives Graph quadratic inequalities Graph linearquadratic and quadratic systems of inequalities Activity 6 Introduction What do satellite dishes, car headlights, and camera lenses have in common?
More informationTeaching Quadratic Functions
Teaching Quadratic Functions Contents Overview Lesson Plans Worksheets A1 to A12: Classroom activities and exercises S1 to S11: Homework worksheets Teaching Quadratic Functions: National Curriculum Content
More informationZero: 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
More informationEquations, 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
More informationSolving Equations of Degree 1 (Linear Equations):
Solving Equations of Degree 1 (Linear Equations): Definition: The equation a b0 is called an equation of degree 1. Eample: Solve for : 0. 0, Or. Dividing both sides by : giving. Easy! Check the answer!
More information6 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
More informationYear 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
More information1.5. 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
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + 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
More informationLesson 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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationSTRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula
UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving
More informationRoots, 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
More informationSection 33 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
More informationThe 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
More informationPPS TI83 Activities Algebra 12 Teacher Notes
PPS TI83 Activities Algebra 12 Teacher Notes It is an expectation in Portland Public Schools that all teachers are using the TI83 graphing calculator in meaningful ways in Algebra 12. This does not
More informationIndiana 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
More informationWest WindsorPlainsboro Regional School District Algebra I Part 2 Grades 912
West WindsorPlainsboro Regional School District Algebra I Part 2 Grades 912 Unit 1: Polynomials and Factoring Course & Grade Level: Algebra I Part 2, 9 12 This unit involves knowledge and skills relative
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationThe 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
More informationSection V: Quadratic Equations and Functions. Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and CompletingtheSquare
Haberman / Kling MTH 95 Section V: Quadratic Equations and Functions Module : Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and CompletingtheSquare DEFINITION: A quadratic equation
More informationStraight Line Graphs. Teachers Teaching with Technology. Scotland T 3. y = mx + c. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technolog (Scotland) Teachers Teaching with Technolog T 3 Scotland Straight Line Graphs = m + c Teachers Teaching with Technolog (Scotland) STRAIGHT LINE GRAPHS Aim The aim of this
More informationKey. Introduction. What is a Quadratic Equation? Better Math Numeracy Basics Algebra  Rearranging and Solving Quadratic Equations.
Key On screen content Narration voiceover Activity Under the Activities heading of the online program Introduction This topic will cover: the definition of a quadratic equation; how to solve a quadratic
More informationPower of the Quadratic Formula
Power of the Quadratic Formula Name 1. Consider the equation y = x 4 8x 2 + 4. It may be a surprise, but we can use the quadratic the quadratic formula to first solve for x 2. Once we know the value of
More information7.7 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
More informationZeros 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
More informationMethod 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
More information3.6. 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
More informationQuadratic Functions. Copyright Cengage Learning. All rights reserved.
Quadratic Functions 4 Copyright Cengage Learning. All rights reserved. Solving by the Quadratic Formula 2 Example 1 Using the quadratic formula Solve the following quadratic equations. Round your answers
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
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 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More information5.1 The Remainder and Factor Theorems; Synthetic Division
5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials
More informationGRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number.
GRE MATH REVIEW #5 Eponents and Radicals Many numbers can be epressed as the product of a number multiplied by itself a number of times. For eample, 16 can be epressed as. Another way to write this is
More information13 Graphs, Equations and Inequalities
13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,
More informationCalculus Card Matching
Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information
More informationMarch 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
More informationis 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
More informationZeros 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
More information2.3. 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
More informationUnderstanding and Applying Quadratic Relationships
Grade: 8 (Algebra & Geometry 1) Duration: 6 Weeks Summary: Students will identify, compute with, and factor polynomial expressions. Students will learn to recognize the special products of polynomials,
More information3.2. 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.
More informationCommon Curriculum Map. Discipline: Math Course: College Algebra
Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a
More informationBiggar 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
More informationTeaching & Learning Plans. Quadratic Equations. Junior Certificate Syllabus
Teaching & Learning Plans Quadratic Equations Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationSolutions 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
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More informationChapter 1 POLYNOMIAL FUNCTIONS
Chapter POLYNOMIAL FUNCTIONS Have you ever wondered how computer graphics software is able to so quickly draw the smooth, lifelike faces that we see in video games and animated movies? Or how in architectural
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationSCHOOL DISTRICT OF THE CHATHAMS CURRICULUM PROFILE
CONTENT AREA(S): Mathematics COURSE/GRADE LEVEL(S): Honors Algebra 2 (10/11) I. Course Overview In Honors Algebra 2, the concept of mathematical function is developed and refined through the study of real
More informationSystems 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
More informationPolynomials. 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
More informationRoots of quadratic equations
CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More information3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes
Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.
More informationAlgebra 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
More informationPreAP Algebra 2 Lesson 21 Solving 2x2 Systems. 3x 2y 22 (6,2) 5x y 28 (1, 3)
Lesson 21 Solving 2x2 Systems Objectives: The students will be able to solve a 2 x 2 system of equations graphically and by substitution, as well as by elimination. Materials: paper, pencil, graphing
More informationMATHS 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
More informationPolynomials. 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
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationMath 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
More informationAlgebra II Semester Exam Review Sheet
Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationMathematics 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
More informationPolynomial 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
More informationSAMPLE. 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
More informationUnit 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
More informationIntegration of Rational Expressions by Partial Fractions
Integration of Rational Epressions by Partial Fractions INTRODUTION: We start with a few definitions rational epression is formed when a polynomial is divided by another polynomial In a proper rational
More informationCLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.
SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real
More information2.5 ZEROS OF POLYNOMIAL FUNCTIONS. Copyright Cengage Learning. All rights reserved.
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.
More informationTo 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
More informationIn this lesson you will learn to find zeros of polynomial functions that are not factorable.
2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has
More informationx 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac
Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square
More informationQUADRATIC EQUATIONS Use with Section 1.4
QUADRATIC EQUATIONS Use with Section 1.4 OBJECTIVES: Solve Quadratic Equations by Factoring Solve Quadratic Equations Using the Zero Product Property Solve Quadratic Equations Using the Quadratic Formula
More informationSymmetry. A graph is symmetric with respect to the yaxis if, for every point (x, y) on the graph, the point (x, y) is also on the graph.
Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions.
More information3.6 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,
More informationSolving Quadratic Equations
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
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
More informationMINI LESSON. Lesson 5b Solving Quadratic Equations
MINI LESSON Lesson 5b Solving Quadratic Equations Lesson Objectives By the end of this lesson, you should be able to: 1. Determine the number and type of solutions to a QUADRATIC EQUATION by graphing 2.
More informationLyman Memorial High School. PreCalculus Prerequisite Packet. Name:
Lyman Memorial High School PreCalculus Prerequisite Packet Name: Dear PreCalculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These
More informationUnit 4 Solving Equations And Inequalities
Mathematics III Frameworks Student Edition Unit 4 Solving Equations And Inequalities 1 st Edition June 010 Unit 4: Page 1 of 6 Table of Contents INTRODUCTION:... 3 Historical Background Learning Task...
More informationActivity 2. Tracing Paper Inequalities. Objective. Introduction. Problem. Exploration
Objective Graph systems of linear inequalities in two variables in the Cartesian coordinate plane Activity 2 Introduction A set of two or more linear equations is called a system of equations. A set of
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationAlgebra. Indiana Standards 1 ST 6 WEEKS
Chapter 1 Lessons Indiana Standards  11 Variables and Expressions  12 Order of Operations and Evaluating Expressions  13 Real Numbers and the Number Line  14 Properties of Real Numbers  15 Adding
More information