Factoring and Algebraic Fractions
|
|
- Martha Booker
- 5 years ago
- Views:
Transcription
1 Worksheet. Algebraic Fractions Section Factoring and Algebraic Fractions As pointed out in worksheet., we can use factoring to simplify algebraic expressions, and in particular we can use it to simplify algebraic fractions. Calculations using algebraic functions are similar to calculations involving fractions. So when adding together fractions with different denominators, we must first find the lowest common multiple. Example : a + b a 5 5(a + b) 5 (a) 5 5a + 5b 4a 0 0 5a + 5b 4a 0 a + 5b 0 Example : x y + y (y + ) y x 9y y + 9y (y + ) 9y y 7y
2 Example : y + + y (y ) (y + )(y ) (y + ) (y )(y + ) y (y + )(y ) y + (y )(y + ) y y (y + )(y ) y 5 (y + )(y ) (y + 5) (y + )(y ) Sometimes it is difficult to find a simple expression that is a multiple of two algebraic expressions. When this is the case it is perfectly acceptable to multiply the two expressions together even though this will not necessarily form the smallest common multiple. You should check at the end of the calculation in the final fraction that there are no common factors in the numerator and denominator; if there are, you can always cancel them to give an equivalent but simpler fraction. Exercises:. Simplify the following algebraic expressions: (a) x + x (b) m m 7 5 (c) 4t + t 5 (d) m+ m 4 (e) m+4 + m 7 y (f) y y+ y+ (g) 5 t+ + 4 t (h) m + 4m m+4 m+5 (i) 4 y+ 5 y+ (j) 7 4x + 5 Section Multiplication and Division As in numerical fractions, the trick with simplifying the multiplication and division of algebraic fractions is to look for common factors both before and after calculation. Once common factors are cancelled out you get an equivalent fraction in its simplest form. Remember that dividing Page
3 by a fraction is the same operation as multiplying by the reciprocal. That is x x x x For example 6 means how many 6ths are in one whole? The answer is 6. Also, an algebraic expression in the numerator or denominator should be treated as if it were in brackets. For instance x + (x + ) x Example : x x 8 x 8 x 8x x 4 Example : 8 x + 7 x x + x (x + ) 7(x + ) Example : x+ 4y x+4 8y x + 4y x + 4 8y x + 8y 4y x + 4 8y(x + ) 4y(x + 4) (x + ) x + 4 Page
4 Example 4 : 4 x x + 4 (x + ) Example 5 : y 6 x y y x Example 6 : ( x x y y ) ( (y )x (y )(x ) ( x +y ) (x )(y ) (x )y (x )(y ) ) x + y (x )(y ) (y x) (x )(y ) Exercises:. Simplify the following algebraic expressions: (a) m 6 5m (b) m 8 5m 0 (c) 6x+ x+ 8 (d) 9 6x 7 (e) 6pq p 5 7 (f) (x+) 5(x+) 8 6 (g) 4x 7 5 (h) m+ m m+ 5 (i) pq 4p p+ (j) 8(x+) (x+) 9 4(x+) Page 4
5 Section Solving Equations Sometimes we are asked to solve an equation for a particular variable. This means that only the variable should be on one side of an equality sign and the other information in the equation should be on the other side. This is similar to solving equations in one variable as in Worksheet.. However, you may end up with an algebraic expression on one side involving other variables rather than just a number. You should attack these questions in the same way as solving equations for one variable. The following examples and exercises use some of the techniques given in sections one and two of this worksheet. Example : x + x + 5 Multiply each side by 5 - this will eliminate the fractions: 5 ( x + x + 5 ) 5 5 x + 5 x (x ) + (x + ) 45 5x 0 + x x x 5 x 5 8 Page 5
6 Example : Solve for x in terms of y. + y + x ( + ) (y + ) (y + ) (y + )x (multiplying both sides by (y + )) + + x + x x(y + ) (factoring to separate the x) x y + Example : Solve for y in terms of x. 4 + x x + y y 6y (4 + x) 6y (x + ) (multiply both sides by 6y ) y y (4 + x) y(x + ) (4 + x) y isolate y (x + ) (4 + x) y (x + ) In the last two steps, we were aiming to make x the subject of the equation. Exercises:. Solve for x: (a) x+8 x 4 5 (b) x+ + x 4 5 (c) (x ) (x+) (d) + x+ x 4 x+ (e) 5 x+ + x+6 4 Page 6
7 . Solve for x in terms of y: (a) 8 (b) 4 x+ y+ (c) 4(y + ) (x + 5) 8 (d) +y +x y (e) Page 7
8 Exercises. Algebraic Fractions. Simplify the following: (a) x + x 4 (b) + 4 (c) x+ (6x + 5) (d) 4 b b (e) x+ y x+ (f) x 4x + 4 x 4 (g) x+ (h) x+ x+ + x+ (i) 4a + a+5 (a+) 7 4 (j) p ( p p + 5p ) 4 6. Simplify the follwing: (a) 4(x+) 5(x ) (b) x +x x+8 x+4 5x (c) 8x 4 x+7 4 (d) y 6y y+5 y+5 y (e) 5m 7 4m+8 m+ m+6 (f) 6p 4p+ 4 Page 8
Unit 9 Study Sheet Rational Expressions and Types of Equations
Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by
More informationBasic Algebra. CAPS Mathematics
Basic Algebra CAPS Mathematics 1 Outcomes for this TOPIC In this TOPIC you will: Revise factorization. LESSON 1. Revise simplification of algebraic fractions. LESSON. Discuss when trinomials can be factorized.
More informationChapter 7 Rational Expressions, Equations, and Functions
Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions
More informationTable of Contents. Module 1
Table of Contents Module 1 1.1 Order of Operations 1.6 Signed Numbers 1. Factorization of Integers 1.7 Further Signed Numbers 1.3 Fractions 1.8 Power Laws 1.4 Fractions and Decimals 1.9 Introduction to
More informationMath 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:
Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.
More information264 CHAPTER 4. FRACTIONS cm in cm cm ft pounds
6 CHAPTER. FRACTIONS 9. 7cm 61. cm 6. 6ft 6. 0in 67. 10cm 69. pounds .. DIVIDING FRACTIONS 6. Dividing Fractions Suppose that you have four pizzas and each of the pizzas has been sliced into eight equal
More informationExpanding brackets and factorising
Chapter 7 Expanding brackets and factorising This chapter will show you how to expand and simplify expressions with brackets solve equations and inequalities involving brackets factorise by removing a
More informationA Level Maths. Induction Booklet CONTENTS
A Level Maths Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations page 4 Chapter 3 Simultaneous equations page 8 Chapter 4 Factors page 10 Chapter 5 Change the subject
More informationCONTENTS CHECK LIST ACCURACY FRACTIONS INDICES SURDS RATIONALISING THE DENOMINATOR SUBSTITUTION
CONTENTS CHECK LIST - - ACCURACY - 4 - FRACTIONS - 6 - INDICES - 9 - SURDS - - RATIONALISING THE DENOMINATOR - 4 - SUBSTITUTION - 5 - REMOVING BRACKETS - 7 - FACTORISING - 8 - COMMON FACTORS - 8 - DIFFERENCE
More informationSIXTH FORM MATHEMATICS A LEVEL INDUCTION BOOKLET SEPTEMBER Name:
SIXTH FORM MATHEMATICS A LEVEL INDUCTION BOOKLET SEPTEMBER 014 Name: INTRODUCTION TO A LEVEL MATHS Thank you for choosing to study Mathematics in the sixth form at Chelsea Academy. In year 1 you will sit
More informationFractions. Review R.7. Dr. Doug Ensley. January 7, Dr. Doug Ensley Review R.7
Review R.7 Dr. Doug Ensley January 7, 2015 Equivalence of fractions As long as c 0, a b = a c b c Equivalence of fractions As long as c 0, a b = a c b c Examples True or False? 10 18 = 2 5 2 9 = 5 9 10
More informationChapter. Algebra techniques. Syllabus Content A Basic Mathematics 10% Basic algebraic techniques and the solution of equations.
Chapter 2 Algebra techniques Syllabus Content A Basic Mathematics 10% Basic algebraic techniques and the solution of equations. Page 1 2.1 What is algebra? In order to extend the usefulness of mathematical
More informationLESSON 8.1 RATIONAL EXPRESSIONS I
LESSON 8. RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I 7 OVERVIEW Here is what you'll learn in this lesson: Multiplying and Dividing a. Determining when a rational expression is undefined Almost
More informationIntermediate Tier - Algebra revision
Intermediate Tier - Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double brackets Substitution Solving equations Finding nth term
More informationAlgebra Revision Guide
Algebra Revision Guide Stage 4 S J Cooper 1st Edition Collection of like terms... Solving simple equations... Factorisation... 6 Inequalities... 7 Graphs... 9 1. The straight line... 9. The quadratic curve...
More informationMAIDSTONE GRAMMAR SCHOOL FOR GIRLS
MAIDSTONE GRAMMAR SCHOOL FOR GIRLS King Edward VI High School DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET CONTENTS Reading List... 3 Section 1: FRACTIONS... 4 Section : EXPANDING...
More informationA. Incorrect! Perform inverse operations to find the solution. B. Correct! Add 1 to both sides of the equation then divide by 2 to get x = 5.
Test-Prep Math - Problem Drill 07: The Multi-Step Equations Question No. 1 of 10 1. Solve: 2x 1 = 9 Question #01 (A) 4 (B) 5 (C) 1/5 (D) -5 (E) 0 B. Correct! Add 1 to both sides of the equation then divide
More informationSection 2.4: Add and Subtract Rational Expressions
CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall
More informationSelf-Directed Course: Transitional Math Module 4: Algebra
Lesson #1: Solving for the Unknown with no Coefficients During this unit, we will be dealing with several terms: Variable a letter that is used to represent an unknown number Coefficient a number placed
More information5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality
5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality Now that we have studied the Addition Property of Equality and the Multiplication Property of Equality, we can solve
More informationP.1 Prerequisite skills Basic Algebra Skills
P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable
More informationMTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE. A rational expression is just a fraction involving polynomials, for example 3x2 2
MTH 1310, SUMMER 2012 DR. GRAHAM-SQUIRE SECTION 1.2: PRECALCULUS REVIEW II Practice: 3, 7, 13, 17, 19, 23, 29, 33, 43, 45, 51, 57, 69, 81, 89 1. Rational Expressions and Other Algebraic Fractions A rational
More informationJUST THE MATHS UNIT NUMBER 1.5. ALGEBRA 5 (Manipulation of algebraic expressions) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.5 ALGEBRA 5 (Manipulation of algebraic expressions) by A.J.Hobson 1.5.1 Simplification of expressions 1.5.2 Factorisation 1.5.3 Completing the square in a quadratic expression
More informationSpring Nikos Apostolakis
Spring 07 Nikos Apostolakis Review of fractions Rational expressions are fractions with numerator and denominator polynomials. We need to remember how we work with fractions (a.k.a. rational numbers) before
More informationNumerical and Algebraic Fractions
Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core
More informationA polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.
LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x
More informationMathematics: Year 12 Transition Work
Mathematics: Year 12 Transition Work There are eight sections for you to study. Each section covers a different skill set. You will work online and on paper. 1. Manipulating directed numbers and substitution
More informationIES Parque Lineal - 2º ESO
UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....
More informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationIntroduction to Algebra
Worksheet 1.9 Introduction to Algebra Section 1 Algebraic Expressions Algebra is a way of writing arithmetic in a general form. You have already come across some algebraic expressions in previous worksheets.
More informationRemember, you may not use a calculator when you take the assessment test.
Elementary Algebra problems you can use for practice. Remember, you may not use a calculator when you take the assessment test. Use these problems to help you get up to speed. Perform the indicated operation.
More informationChapter 5 Rational Expressions
Worksheet 4 (5.1 Chapter 5 Rational Expressions 5.1 Simplifying Rational Expressions Summary 1: Definitions and General Properties of Rational Numbers and Rational Expressions A rational number can be
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationA constant is a value that is always the same. (This means that the value is constant / unchanging). o
Math 8 Unit 7 Algebra and Graphing Relations Solving Equations Using Models We will be using algebra tiles to help us solve equations. We will practice showing work appropriately symbolically and pictorially
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationIntroduction to A-Level Maths (Bridging Unit)
Introduction to A-Level Maths (Bridging Unit) What is infinity + infinity? To infinity and beyond! SUMMER 017 Tuford Academy Faculty of Mathematics 1 INTRODUCTION TO A LEVEL MATHS AT TUXFORD ACADEMY Thank
More informationINTRODUCTION TO FRACTIONS
INTRODUCTION TO FRACTIONS MEANING AND PROPERTIES OF FRACTIONS Fractions are used to represent parts of a whole. Example: What is the fraction of the shaded area? one-half one-quarter three-eighths 4 The
More informationFactorizing Algebraic Expressions
1 of 60 Factorizing Algebraic Expressions 2 of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. Expanding or multiplying out a(b + c) ab + ac Factorizing Often: When
More informationFactoring Review. Rational Expression: A single variable rational expression is an algebraic fraction in which
Factoring Review Factoring methods for factoring polynomial expressions: i) greatest common factor ii) difference of squares iii) factoring trinomials by inspection iv) factoring trinomials by decomposition,
More informationA-LEVEL MATHS Bridging Work 2017
A-LEVEL MATHS Bridging Work 017 Name: Firstly, CONGRATULATIONS for choosing the best A-Level subject there is. A-Level Maths at Wales is not only interesting and enjoyable but is highly regarded by colleges,
More informationMaths Department. A Level Induction Booklet
Maths Department A Level Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations 4 Chapter 3 Simultaneous equations 8 Chapter 4 Factors 10 Chapter 5 Change the subject of the
More informationSail into Summer with Math!
Sail into Summer with Math! For Students Entering Algebra 1 This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review grade level math
More informationChapter 1.6. Perform Operations with Complex Numbers
Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each
More informationMaths Department. A Level Induction Booklet
Maths Department A Level Induction Booklet One of the most important things if you are to succeed at A Level Maths is to ensure you understand all the algebra you met at GCSE. Working through the eamples
More informationMAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS
MAIDSTONE GRAMMAR SCHOOL FOR GIRLS DEPARTMENT OF MATHEMATICS Introduction to A level Maths INDUCTION BOOKLET INTRODUCTION TO A LEVEL MATHS AT MGGS Thank you for choosing to study Mathematics in the sixth
More informationREAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?
REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate
More informationSection 1.1 Notes. Real Numbers
Section 1.1 Notes Real Numbers 1 Types of Real Numbers The Natural Numbers 1,,, 4, 5, 6,... These are also sometimes called counting numbers. Denoted by the symbol N Integers..., 6, 5, 4,,, 1, 0, 1,,,
More informationBASICS OF ALGEBRA M.K. HOME TUITION. Mathematics Revision Guides. Level: GCSE Foundation Tier
Mathematics Revision Guides Basics of Algebra Page 1 of 7 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF ALGEBRA Version: 3.1 Date: 01-12-2013 Mathematics Revision
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
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 informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More information1h4 Exponents date: Remember: never leave negative exponents in your final answers. Exponent rules:
h4 Exponents date: Remember: never leave negative exponents in your final answers. Exponent rules:. x m x n = x m+n when multiplying powers with the same base: keep the base and add the exponents. x m
More informationJUST THE MATHS UNIT NUMBER 1.9. ALGEBRA 9 (The theory of partial fractions) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1. ALGEBRA (The theory of partial fractions) by A.J.Hobson 1..1 Introduction 1..2 Standard types of partial fraction problem 1.. Exercises 1..4 Answers to exercises UNIT 1. -
More informationAlgebra 1 Summer Assignment 2018
Algebra 1 Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Algebra 1 this coming school year. You are advised to be familiar
More informationA field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
More informationCHAPTER 1. Review of Algebra
CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (although some of the exercises are harder). Some of it particularly the work on logarithms may be new if you
More informationFunctions and Their Graphs
Functions and Their Graphs DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x) Y to each element x D. A symbolic way to say y is a function of x
More informationExponents Drill. Warm-up Problems. Problem 1 If (x 3 y 3 ) -3 = (xy) -z, what is z? A) -6 B) 0 C) 1 D) 6 E) 9. Problem 2 36 =?
Exponents Drill Warm-up Problems Problem 1 If (x 3 y 3 ) -3 = (xy) -z, what is z? A) -6 B) 0 C) 1 D) 6 E) 9 Problem 2 3 36 4 4 3 2 =? A) 0 B) 1/36 C) 1/6 D) 6 E) 36 Problem 3 3 ( xy) =? 6 6 x y A) (xy)
More informationLesson 7: Watch Your Step!
In previous lessons, we have looked at techniques for solving equations, a common theme throughout algebra. In this lesson, we examine some potential dangers where our intuition about algebra may need
More informationCore Mathematics 3 Algebra
http://kumarmathsweeblycom/ Core Mathematics 3 Algebra Edited by K V Kumaran Core Maths 3 Algebra Page Algebra fractions C3 The specifications suggest that you should be able to do the following: Simplify
More informationAlgebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review
Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,
More information1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.
1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors
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 informationPLEASE NOTE THAT YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!
DETAILED SOLUTIONS AND CONCEPTS - INTRODUCTION TO ALGEBRA 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 informationPreparing for A-Level Mathematics Summer 2017
Preparing for A-Level Mathematics Summer 017 INTRODUCTION TO A LEVEL MATHS Thank you for choosing to study Mathematics in the sith form. You will sit two modules in Pure Mathematics (C1 and C) as well
More informationChapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
More informationSection 4.6 Negative Exponents
Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals
More informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE ALGEBRA Module Topics Simplifying expressions and algebraic functions Rearranging formulae Indices 4 Rationalising a denominator
More informationDivisibility, Factors, and Multiples
Divisibility, Factors, and Multiples An Integer is said to have divisibility with another non-zero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number
More information6-3 Solving Systems by Elimination
Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two
More informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
More informationSection 3 1C: Solving a System of Equations by Elimination
Section 3 1C: Solving a System of Equations by Elimination Solve each system by the elimination method. Example 1 Example 2 3x + 5y = 2 3x + y = 14 3x 2y = 3 8x + 2y = 30 Add and Add and 3x + 5y = 2 3x
More informationADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET
NAME ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET Part I. Order of Operations (PEMDAS) Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division. Addition & Subtraction. Tutorial:
More informationFactorisation CHAPTER Introduction
FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write
More informationAlgebra. Mathematics Help Sheet. The University of Sydney Business School
Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,
More informationInverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
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 informationNot drawn accurately
Q1. A trapezium has parallel sides of length (x + 1) cm and (x + 2) cm. The perpendicular distance between the parallel sides is x cm. The area of the trapezium is 10 cm 2. Not drawn accurately Find the
More informationCore Mathematics 2 Algebra
Core Mathematics 2 Algebra Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Algebra 1 Algebra and functions Simple algebraic division; use of the Factor Theorem and the Remainder Theorem.
More information4.5 Multiplication and Division of Rational Expressions
.5. Multiplication and Division of Rational Epressions www.ck2.org.5 Multiplication and Division of Rational Epressions Learning Objectives Multiply rational epressions involving monomials. Multiply rational
More informationYOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
More informationChapter 1 A Survey of Divisibility 14
Chapter 1 A Survey of Divisibility 14 SECTION C Euclidean Algorithm By the end of this section you will be able to use properties of the greatest common divisor (gcd) obtain the gcd using the Euclidean
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationMath 154 :: Elementary Algebra
Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with
More informationGeometry Summer Assignment
2018-2019 Geometry Summer Assignment You must show all work to earn full credit. This assignment will be due Friday, August 24, 2018. It will be worth 50 points. All of these skills are necessary to be
More informationSupplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Section 16 Solving Single Step Equations
Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 2 Please watch Section 16 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item67.cfm
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationAlgebra is a part of mathematics in which numbers and letters are used. Numbers and letters are combined by the arithmetic operations.
Colegio Herma. Maths. Bilingual Department by Isabel Martos Martínez. 2013 WHAT IS ALGEBRA? Algebra is a part of mathematics in which numbers and letters are used. Numbers and letters are combined by the
More informationAlgebra 31 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationMathematics Higher Tier, June /1H (Paper 1, non-calculator)
Link to past paper on AQA website: www.aqa.org.uk The associated question paper is available to download freely from the AQA website. To navigate around the website, choose QUALIFICATIONS, GCSE, MATHS,
More informationMATCHING. Match the correct vocabulary word with its definition
Name Algebra I Block UNIT 2 STUDY GUIDE Ms. Metzger MATCHING. Match the correct vocabulary word with its definition 1. Whole Numbers 2. Integers A. A value for a variable that makes an equation true B.
More informationPROBLEMS ON DIGITS / NUMBERS / FRACTIONS Ex-1: The sum of five consecutive positive integers is 55. The sum of the squares of the extreme terms is (1) 308 (2) 240 (3) 250 (4) 180 Let the five consecutive
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 informationSection 3-4: Least Common Multiple and Greatest Common Factor
Section -: Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
More informationMathematics 96 (3581) CA (Class Addendum) 1: Commutativity Mt. San Jacinto College Menifee Valley Campus Spring 2013
Mathematics 96 (3581) CA (Class Addendum) 1: Commutativity Mt. San Jacinto College Menifee Valley Campus Spring 2013 Name This class handout is worth a maximum of five (5) points. It is due no later than
More information2Algebraic. foundations
2Algebraic foundations 2. Kick off with CAS 2.2 Algebraic skills 2.3 Pascal s triangle and binomial expansions 2.4 The Binomial theorem 2.5 Sets of real numbers 2.6 Surds 2.7 Review c02algebraicfoundations.indd
More informationUnit 3 Factors & Products
1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 2006 Director Sarah R. Martinez,
More informationx y = 2 x + 2y = 14 x = 2, y = 0 x = 3, y = 1 x = 4, y = 2 x = 5, y = 3 x = 6, y = 4 x = 7, y = 5 x = 0, y = 7 x = 2, y = 6 x = 4, y = 5
List six positive integer solutions for each of these equations and comment on your results. Two have been done for you. x y = x + y = 4 x =, y = 0 x = 3, y = x = 4, y = x = 5, y = 3 x = 6, y = 4 x = 7,
More informationWeek 1 Factor & Remainder Past Paper Questions - SOLUTIONS
Question Solution Marks C2 JUNE 2010 Q2 f(3) = 3(3) 3 5(3) 2 58(3) + 40 f(3) = 98 So, the remainder is 98 Substitute x = 3 into 98 (If it is not stated that 98 is the remainder, do not give this C2 JUNE
More information