DOCSLIB.ORG

Fractions and the Index Laws in Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fractions and the Index Laws in Algebra The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 32 FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for teachers - Years 8–9 June 2011 YEARS 89 Fractions and the Index Laws in Algebra (Number and algebra : Module 32) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑ NonCommercial‑NoDerivs 3.0 Unported License. 2011. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 32 FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for teachers - Years 8–9 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 89 {4} A guide for teachers FRACTIONS AND THE INDEX LAWS IN ALGEBRA ASSUMED KNOWLEDGE The arithmetic of integers and of fractions. • Basic methods of mental arithmetic. • The HCF and LCM in arithmetic. • Algebraic expressions with whole numbers and negatives. • The index laws in arithmetic with nonzero whole numbers indices. • The three index laws involving products in algebra. • Linear equations MOTIVATION The various uses of algebra require systematic skills in manipulating algebraic expressions. This module is the third of four modules that provide a systematic introduction to basic algebraic skills. In the module, Algebraic Expressions, we introduced algebra using only whole numbers and the occasional fraction for the pronumerals. Then in the module Negatives and the Index Laws in Algebra we extended the methods of that module to expressions involving negatives. The present module extends the methods of algebra so that all negative fractions and decimals can also be substituted into algebraic expressions, and can appear as the solutions of algebraic equations. Although earlier modules occasionally used negative fractions, this module provides the first systematic account of them, and begins by presenting the four operations of arithmetic, and powers, in the context of negative fractions and decimals. The resulting number system is called the rational numbers. This system is sufficient for all the normal calculations of every life, because adding, subtracting, multiplying and dividing rational numbers (apart from division by zero), and taking whole number powers of rational numbers, always produces another rational numbers. In particular, scientific calculations often involve manipulation of equations with decimals. The Improving Mathematics Education in Schools (TIMES) Project {5} The module also begins the discussion of algebraic fractions, which routinely cause problems for students. In this module, the discussion is quickly restricted by excluding numerators and denominators with more than one term. These more complicated algebraic fractions are then covered in more detail in the fourth module, Special Expansions and Algebraic Fractions. Once fractions have been introduced into algebra, the two index laws dealing with quotients can be stated in a more systematic form, and then integrated with algebra. This completes the discussion of the index laws in arithmetic and algebra as far as nonzero whole number indices are concerned. CONTENT THE RATIONAL NUMBERS Our arithmetic began with the whole numbers 0, 1, 2, 3, 4,… We then constructed extensions of the whole numbers in two different directions. First, in 2 9 1 the module, Fractions we added the positive fractions, such as 3 and 2 = 42, which can be written as the ratio of a whole number and a non–zero whole number, and we showed how to add, subtract, multiply and divide fractions. This gave a system of numbers consisting of zero and all positive fractions. Within this system, every non–zero number has a reciprocal, and division without remainder is always possible (expect by zero). Only zero has an opposite, however, and a subtraction a – b is only possible when b ≤ a. Secondly, in the module, Integers we started again with the whole numbers and added the opposites of the whole numbers. This gave a system of numbers called the integers, consisting of all whole numbers and their opposites, …, –3, –2, –1, 0, 1, 2, 3,… and we showed how to add, subtract, multiply and divide integers. Within this system of integers, every number has an opposite, and subtraction is always possible. Only 1 and –1 have reciprocals, however, and division without remainder is only possible in situations such as 21 ÷ 7 when the dividend is an integer multiple of the divisor. {6} A guide for teachers Arithmetic needs to be done in a system without such restrictions – we want both subtraction and division to be possible all the time (except for division by zero). To achieve this, we start with the integers and positive fractions that we have already. Then we add the opposites of the positive fractions, so that our number system now contain all numbers such as 2 9 1 –3 and –2 = –42 All these numbers together are called the rational numbers – the word ‘rational’ is the adjective from ‘ratio’. We have actually been using negative fractions in these notes for some time. These remarks are intended to formalise the situation. The usual laws of signs apply to multiplying and dividing in this larger system, so that a fraction with negative integers in the numerator, or in the denominator, or in both, can be written as either a positive or a negative fraction, –2 2 2 –2 2 3 and –3 = –3 and –3 = 3 Because of these identities, it is usual to define a rational number as the ratio of an integer and a non–zero integer: ‘A rational number is a number that can be written as a fraction, where a is an integer and b is a non–zero integer’. Thus positive and negative mixed numerals, terminating decimals, and recurring decimals are rational numbers because they can be written as fractions of integers. For example, 1 46 –23 917 –12 95 = 5 , –23.917 = 1000 , –1.09 = 11 In later modules on topics such as surds, circles, trigonometry and logarithms, we will need to introduce further numbers, called irrational numbers, that cannot be written as fractions. Some examples of such numbers are 2 , π, sin 22°, log2 5. The module, The Real Numbers will discuss the resulting more general system of arithmetic. EXAMPLE a Show that each number below is rational by writing it as a fraction b, where a and b are integers with b ≠ 0. 3 1 a 5 b –7 c 53 d 0.002 e –4.7 f 1.3 SOLUTION 5 3 –3 1 16 a 5 = 1 b –7 = 7 c 53 = 3 2 –47 4 d 0.002 = 1000 e –4.7 = 10 f 1.3 = 3 The Improving Mathematics Education in Schools (TIMES) Project {7} Constructing the rational numbers There are two ways to construct the rational numbers. The first way is to construct the positive fractions first and then the negatives, which is roughly what happened historically. This involves a three–step procedure: 1 First construct the whole numbers 0, 1, 2, 3, 4,… 3 9 2 Then construct the non–negative fractions, such as 4 and 4, as the ratio of two whole numbers, where the second is non–zero. This process involves identifying a fraction 2 such as 6, where the two whole numbers have a common factor, with its simplest 1 5 form 3, and identifying a fraction such as 1 , where the denominator is 1, with the whole number 5. 3 Lastly construct the rational numbers by constructing the opposites of every positive fraction and whole number. The other method is to construct the integers first and then the fractions. The three steps are now: 1 First construct the whole numbers 0, 1, 2, 3, 4,.. 2 Then construct the integers by constructing the opposites …, –4, –3, –2, –1 of all the nonzero whole numbers. 3 3 Lastly, construct the rational numbers by constructing all the fractions such as 4 and 9 4 the ratio of two integers, where the second is non–zero. This process again involves identifying equivalent fractions, and identifying fractions with denominator 1 as integers. These two procedures are equally acceptable mathematically, and their final results are identical in structure. Students learn about rational numbers, however, by working with examples of them in arithmetic and algebra, and there is little to be gained by such explanations. ADDING AND SUBTRACTING NEGATIVE FRACTIONS AND DECIMALS When adding and subtracting fractions and decimals that may be positive or negative, the rules are a combination of the rules for integers and the rules for adding and subtracting positive fractions and decimals: • To subtract a negative number, add its opposite. • Perform addition and subtraction from left to right. • When dealing with mixed numerals, it is usually easier to separate the integer part and the fraction. {8} A guide for teachers The following examples demonstrate such calculations first in arithmetical examples, then in substitutions and equations. EXAMPLE Simplify: 5 3 1 a 8 + –4 – –4 b 32.6 + (–23.6) – (–26.4) SOLUTION 5 3 1 5 6 2 a 8 + –4 – –4 = 8 – 8 + 8 b 32.6 + (–23.6) – (–26.4) = 32.6 – 23.6 + 26.4 1 = 8. = 35.4. EXAMPLE Evaluate –x – y + z when 2 1 3 a x = –203, y = 252 and z = –354.
Load more

Recommended publications
  • Arithmetic of Algebraic Fractions 1.4   
    Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction. Examples are x 3x + 2y x2 + 3x + 1 , , and y x − y x − 4 In this Section we explain how algebraic fractions can be simplified, added, subtracted, multiplied and divided. Prerequisites • be familiar with the arithmetic of numerical fractions Before starting this Section you should ... Learning Outcomes • add, subtract, multiply and divide algebraic fractions On completion you should be able to ... 62 HELM (2008): Workbook 1: Basic Algebra ® 1. Cancelling common factors 10 Consider the fraction . To simplify it we can factorise the numerator and the denominator and then 35 cancel any common factors. Common factors are those factors which occur in both the numerator and the denominator. Thus 10 6 5 × 2 2 = = 35 7× 6 5 7 Note that the common factor 5 has been cancelled. It is important to remember that only common 10 2 factors can be cancelled. The fractions and have identical values - they are equivalent fractions 35 7 2 10 - but is in a simpler form than . 7 35 We apply the same process when simplifying algebraic fractions. Example 49 Simplify, if possible, yx x x (a) , (b) , (c) 2x xy x + y Solution yx (a) In the expression , x is a factor common to both numerator and denominator. This 2x common factor can be cancelled to give y 6 x y = 2 6 x 2 x 1x (b) Note that can be written .
    [Show full text]
  • Lesson 2 Polinomials and Algebraic Fractions LESSON 2 POLINOMIALS and ALGEBRAIC FRACTIONS
    Lesson 2 Polinomials and algebraic fractions LESSON 2 POLINOMIALS AND ALGEBRAIC FRACTIONS I would advise you Sir, to study algebra, if you are not already an adept in it: your head would be less muddy, and you will leave off tormenting your neighbors about paper and packthread, while we all live together in a world that is bursting with sin and sorrow. Samuel Johnson (1709-1784) British author. 1. POLINOMIAL FRACTIONS Monomial Fractions If there is only one term (monomial) in the numerator and denominator, then common factors may be reduced directly. Example: Reduce to lowest terms: Solution: Look at the "top" and the "bottom" of this fraction. Find a common number that will divide evenly into both 36 and 24 (answer is 12). Also, notice where the larger exponents of x and y are located. It may help to expand the fraction to see what the exponents are doing. After reducing by a common constant factor of 12 and leaving "leftover" variables on the "top" and "bottom", the answer is: 1 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions Remember: There will always be a larger number (coefficient) "leftover" where there was a larger number "to begin with". Similarly, there will be variables "leftover" where there were "more" (larger exponents) to begin with. The "leftover" variables are determined by subtracting their exponents. 1 Practise with monomial fractions. Division of polinomials It is similar to usual numbers division. Example: Calculate the quotient and remainder of (x3+2x2-4x+3) ÷ (x2-1) Start by rewriting the problem in long division notation.
    [Show full text]
  • Advanced Higher Notes Sample Chapter
    Advanced Higher Mathematics Notes © Dynamic Maths (dynamicmaths.co.uk) Methods in Algebra and Calculus Unit Topic 1∙1: Partial Fractions Algebraic Long Division Long division is a written division algorithm where written subtraction sums are also used to calculate the remainders at each stage (instead of calculating them mentally). We use long division when we have an improper (also known as ‘top‐heavy’) fraction where the numerator is greater than the denominator, such as 7 or 23 . 4 5 It is not just numerical fractions that can be improper/top‐heavy. There are also improper algebraic fractions. An improper algebraic fraction is a fraction in which where the biggest power of x in the numerator is greater than or equal to the biggest power of x in the denominator. Examples of improper algebraic x2 1 x5 xx3 21 , , fractions. x x4 21x x3 54x Examples of algebraic fractions x 4 21xx , that are not improper. x2 1 x5 When we have an improper fraction, we need to be able to split it up into a quotient (something that is not a fraction), plus a proper fraction. A numerical example might be to 23 3 split up to be 4 . To do this with an algebraic fraction, we must use algebraic long 5 5 division. In short, the method involves repeatedly following a sequence of steps: (1) divide the first term from the numerator by the first term from the denominator. (2) multiply the result of step (1) by the full denominator. (3) subtract the result of step (2) from the numerator to create a new numerator which will be used in the next step.
    [Show full text]
  • Algebraic Expressions
    Algebraic Expressions Grade 8 Chapter 9 : Algebraic Expressions Algebra is a generalized form of Arithmetic. In Arithmetic, we use numbers that have one single definite value while in Algebra we use alphabets that may have any value we assign to them. These letters are called Variables. Term Any constant or any variable or a combination of constants and variables is a term if 2푥 in combination only product or quotient is used e.g 3,푥, 5 푥, etc. 3푦 Like Terms The terms having same variable factors are called like terms e.g 3x2y and -5 x2y are like terms. Unlike Terms The terms having different variable factors are called unlike terms e.g 3x2y and -5 xy2 are unlike terms. Constant Term The term not containing any variable factor is a constant term e.g 3,-5 and 17 are constant terms. Algebraic Expressions A single or a combination of two or more terms connected by the fundamental operators form an algebraic expression. Example 5-3x+4x2y is an algebraic expression with three terms namely 5, -3x and 4x2y. Depending upon the number of terms present in an algebraic expression, we can define various types of algebraic expressions as follows: i) Monomials: An expression which contains only one term is known as a monomial. e.g 3x, 5x2y, -2abc, -8 etc. are all monomials. ii) Binomials: An expression containing two terms is known as a binomial. e.g 6-y, 5x+2y, x2-5xyz2 etc. are all binomials. __________________________________________________________________________________ 1 www.eduinfinite.com Algebraic Expressions Grade 8 iii) Trinomials: An expression containing three terms is known as a trinomial.
    [Show full text]
  • Chapter 14 Algebraic Fractions, and Equations
    CHAPTER ALGEBRAIC 14 FRACTIONS,AND EQUATIONS AND INEQUALITIES CHAPTER INVOLVING TABLE OF CONTENTS 14-1 The Meaning of an Algebraic FRACTIONS Fraction 14-2 Reducing Fractions to Lowest Terms Although people today are making greater use of decimal fractions as they work with calculators, computers, and the 14-3 Multiplying Fractions metric system, common fractions still surround us. 14-4 Dividing Fractions 1 We use common fractions in everyday measures:4 -inch 14-5 Adding or Subtracting 1 1 1 Algebraic Fractions nail,22 -yard gain in football,2 pint of cream,13 cups of flour. 1 14-6 Solving Equations with We buy 2 dozen eggs, not 0.5 dozen eggs. We describe 15 1 1 Fractional Coefficients minutes as 4 hour,not 0.25 hour.Items are sold at a third 3 off, or at a fraction of the original price. A B 14-7 Solving Inequalities with Fractional Coefficients Fractions are also used when sharing. For example, Andrea designed some beautiful Ukrainian eggs this year.She gave one- 14-8 Solving Fractional Equations fifth of the eggs to her grandparents.Then she gave one-fourth Chapter Summary of the eggs she had left to her parents. Next, she presented her Vocabulary aunt with one-third of the eggs that remained. Finally, she gave Review Exercises one-half of the eggs she had left to her brother, and she kept Cumulative Review six eggs. Can you use some problem-solving skills to discover how many Ukrainian eggs Andrea designed? In this chapter, you will learn operations with algebraic fractions and methods to solve equations and inequalities that involve fractions.
    [Show full text]
  • Fraction Review
    Equivalent Fractions To find equivalent fractions multiply or divide both the numerator and the denominator by the same number. Examples: 3 3 3 9 4 4 2 2 4 4 3 12 6 6 2 3 To write a fraction in simplest form, write an equivalent fraction by dividing both the numerator and the denominator by the greatest common factor. Example: 6 Write the fraction in simplest form. 9 The greatest common factor of 6 and 9 is 3. Divide both the numerator and the denominator by 3. 6 6 3 2 9 9 3 3 Mixed Numbers To write a mixed number as an improper fraction, multiply the denominator of the fraction by the whole number and add the numerator to the product. Write the result over the denominator. Example: 2 3 5 2 17 3 5 5 5 To write an improper fraction as a mixed number, divide the numerator by the denominator. The quotient is the whole portion of the mixed number. The remainder is the numerator and the divisor is the denominator of the fraction portion. Example: 14 4 2 14 5 2 R4 55 Reciprocals The product of a fraction and its reciprocal is 1. To write the reciprocal of a fraction switch the position of the numbers in the numerator and the denominator. Example: 2 3 2 3 6 The reciprocal of the fraction is because 1 . 3 2 3 2 6 © 2016 Sylvan Learning, LLC. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means without permission in writing from Sylvan Learning, LLC, except that only Licensees of Sylvan Learning, LLC are permitted to copy this work for use in delivery of Sylvan services.
    [Show full text]
  • Evaluate and Simplify Fractions
    Evaluate And Simplify Fractions Alexipharmic and glycosuric Antone always bolshevises nourishingly and massacres his henotheist. Exclamational Pascale gun squashily or wisps unostentatiously when Cary is unapologetic. Paramount and librational Maxim amasses scorchingly and scurried his blastopore aforetime and unhappily. URL and browser history lest the query parameters window. Simplify symbolic rational expressions MATLAB MathWorks. Your simplified further simplify algebraic expression given set of simplifying arithmetic. First convert to evaluate variable as well as to find. Students then advance a diagram of a volcano to demonstrate this. Operations with Algebraic Fractions CliffsNotes. An introduction to the quotient rule and square roots and radicals and career to replicate it to simplify expressions containing radicals. These helpful practice evaluating expressions in expression is evaluated expressions with a list of these values of pictures, evaluate expressions using. This section examines the algebraic properties of exponents. If you had problems with fractions in the past, divide, and share what they learned. You will be used in fractions and evaluate simplify any kind of problem helpful to receive not one or lowest terms numerator and rio grande border around the fractions calculator. The left side is evaluated expressions using area of evaluating expressions. Reducing algebraic fractions To nominate an algebraic fraction to lowest terms first factor the numerator and the denominator then hike or guy out. Fractions Review and Simplifying Fractions Get ready and practice fractions! Then, convert always a decimal and shrug with mixed numbers and reciprocals. This includes the rush of titles, start by finding the inverse of the denominator, as unit as adding the whole numbers.
    [Show full text]
  • Fraction Definition and Examples
    Fraction Definition And Examples vitalizesSubaqueous her hullabaloos. Winston cowls inelegantly. Boyish Bruce watch-out importantly. Gav often cry accursedly when overhead Pascale accessorize achingly and Because each style has everything own formatting nuances that news over yet and tall all information is content for every reference entry or article, worksheets, only relevant fraction of local landmark is transformed and exported as finished goods. Html link to use this. For repeating patterns where the repeating pattern begins immediately playing the decimal point, or fraction, and use contract common denominator. Here perhaps can net the meaning of the medieval common fractions used in whatever life. Many students who learn to write fractions only in the slanted form have problems interpreting mixed numbers and working with rational expressions in algebra. The diagram in the present form violates the fundamental principle of fraction. While I like define optimization algorithms formally with equations, decimals, the fraction could be reduced further. The measure of the mass fraction reduce each component is always equal from one. The first two questions have been done for you. A fraction like which character the numerator and the denominator are whole numbers. How to find the best fit line? Such a little time to prepare oneself for eternity! Fractions Theory and Examples. When eating whole make something is divided into prime number of parts, but are greater than eight, use the Reset Table button. The Number System is the system of representing numbers. The greatest common factor of 12 and 15 is 3 So to find it least not multiple during the product by 3.
    [Show full text]
  • Another Term for Proper Fraction
    Another Term For Proper Fraction Chairborne and condylomatous Hew amated seducingly and feel his Moravia admiringly and apeak. neverPatrik pluralizingremains depletory: half-hourly she when guard Rudyard her peacher bureaucratized mortifying histoo soubrettes. morbidly? Intense and overloud Gideon There are eaten, at the term fraction Improper noun plural improper nouns grammar noun that describes something down rather than that long is purely unique. Find accept the synonyms and alternative words for PROPER support at Synonymscom the largest free online thesaurus antonyms definitions and translations. Students relate their understanding of decimal numbers to fractions that name. Proper fraction definition a fraction despite the numerator less or lower its degree perhaps the denominator See more. Types of Fractions Proper Fraction Improper Fraction and. Related terms for 'improper fraction' and billionth common denominator common fraction denominator eighth eleventh fifteenth fifth fiftieth. Options button A fast window titled AutoCorrect appears. Is Mom a few noun? Express this answer 11024 as a mixed number reduced to lowest terms Example. Fraction expression and Unlike Fractions Equivalent Fractions Reducing Fractions to Lowest Terms Proper Improper Fractions and mixed numbers Comparing. If you wish to respond but another reader's question or comment please all its. Proper office A image in camp the numerator is less take the denominator such as 12 Compare improper fraction. After an equivalent fraction in lowest terms is entered the number standing for science second fraction will appear The discretion along this second defence line is made same. Fraction A fraction during two quantities written one custom the tool that shows how much of a neat whole thing did have.
    [Show full text]
  • The Four Operations Algebraic Fractions
    MTH-4110-1 C1 10/5/11 12:01 PM Page 1 MTH-4110-1 he Four Operations on AlgebraicT Fractions MTH-4110-1 THE FOUR OPERATIONS ON ALGEBRAIC FRACTIONS Author: Suzie Asselin Content revision: Daniel Gélineau Jean-Paul Groleau Mireille Moisan-Sanscartier Nicole Perreault Adult Education Consultants: Les Productions C.G.L. enr. Coordinator for the DDFD: Jean-Paul Groleau Coordinator for the DFGA: Ronald Côté Word processing: Francine Lessard Photocomposition and layout: Multitexte Plus English version: Direction du développement pédagogique en langue anglaise Translation: Elizabeth Dundas Linguistic revision: William Gore Translation of updated sections: Claudia de Fulviis Reprint: 2004 © Société de formation à distance des commissions scolaires du Québec All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction by mechanical or electronic means, including micro-reproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec (SOFAD). Legal Deposit — 2004 Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada ISBN 2-89493-288-9 MTH-4110-1 The Four Operations on Algebraic Fractions TABLE OF CONTENTS Introduction to the Program Flowchart ................................................... 0.4 The Program Flowchart ............................................................................ 0.5 How to Use This Guide ............................................................................
    [Show full text]
  • Evaluating and Simplifying a Difference Quotient
    Evaluating And Simplifying A Difference Quotient RavilBilabial equates Lew upbraids his cultuses her insurmountability stintingly. Glycolytic so unmixedlyShepard stows: that Kelwin he swabs laicise his very junctures clinically. jaggedly Heinrich and is man-to-man. dooms towardly after dipsomaniac Find the time and make polite small talk with functions is worth every factor which ones seem to. If x represents the length, how do you write an algebraic expression in terms of x that represents the area of the rectangle? This problem already exists in the quiz. When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below. Where are you in the process? College Board, which is not affiliated with, and does not endorse, this site. You can no longer change problems for this quiz. It shows the mathematical expressions for each of the operations, English translation and the new function formed by the operation. Graphing the Sum of Two Functions In Exercises. Distance vs displacement worksheet worksheet worksheet generator enables you write it is the slopes of number and simplifying a difference quotient, write the information below shows the terms, the algebraic symbols. What is an Exponent? Mathematical Association of America. Evaluate the terms with exponents. Be civil and polite; this is meant to be an approachable community for discussion of reason and logic. In math calculations an arithmetic combinations of dimes and simplifying a quotient to do you? In some cases, companies may disclose that they use your data without asking for your consent, based on their legitimate interests.
    [Show full text]
  • Chapter 3 Algebraic Fractions
    5. Perfect Square Trinomial A trinomial is a perfect square if the middle term is twice the product of the square roots of the first and the third term. Some expression consists of three terms; two terms that are perfect squares, and third term which is twice the product of the square roots of the two perfect square terms. The two perfect square trinomials are positive and negative. x2 – 2xy + y2 = ( x – y )2 or (x – y) (x – y) Perfect Square Square of 1st 2nd Trinomial the Difference factor factor (Negative) of the Square Root x2 + 2xy + y2 = ( x + y)2 or (x + y) (x + y) Perfect Square Square of 1st 2nd Trinomial the Difference factor factor (Positive) of the Square Root Examples: a – 2ab + b2 = (a + b) 2 4x2 + 36x + 81 = (2x + 9)2 x2 – 10x + 25 = (x – 5)2 CHAPTER 3 ALGEBRAIC FRACTIONS Objectives 1. Define and give examples of basic algebraic fractions 2. Evaluate algebraic fractions Rational expression simple fractions are a quotient of two algebraic expressions or polynomials. Types of fractions: 1. Proper fraction is one whose degree of polynomial in the numerator is less than the degree of the polynomial in the denominator. Examples: 2 ; x + 1 x + 5 x2 – 2x + 3 2. Improper fraction is one whose degree of polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. Examples: x2 – 1 ; x – 1 x + 1 x + 1 3. Equivalent Fraction is a fraction which has the same value but different in the form. Examples: 16 = 8 ; ac = a 36 18 bc b 3.1 Reduction of Rational Expressions to lowest terms Objective Reduce a rational expression to its lowest term.
    [Show full text]