We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byMiles Mosey
Modified about 1 year ago
Slide 7- 1 Copyright © 2012 Pearson Education, Inc.
6.2 Rational Expressions and Functions: Adding and Subtracting ■ When Denominators Are the Same ■ When Denominators Are Different
Slide 6- 3 Copyright © 2012 Pearson Education, Inc. Addition and Subtraction with Like Denominators To add or subtract when the denominators are the same, add or subtract the numerators and keep the same denominator.
Slide 6- 4 Copyright © 2012 Pearson Education, Inc. Example Add. Simplify the result, if possible. a)b) c)d) Solution a) b) The denominators are alike, so we add the numerators.
Slide 6- 5 Copyright © 2012 Pearson Education, Inc. Example continued c) d) Factoring Combining like terms Combining like terms in the numerator
Slide 6- 6 Copyright © 2012 Pearson Education, Inc. Example Subtract and, if possible, simplify: a)b) Solution a) The parentheses are needed to make sure that we subtract both terms. Removing the parentheses and changing the signs (using the distributive law) Combining like terms
Slide 6- 7 Copyright © 2012 Pearson Education, Inc. Example continued b) Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying Removing a factor equal to 1
Slide 6- 8 Copyright © 2012 Pearson Education, Inc. Least Common Multiples and Denominators To add or subtract rational expressions that have different denominators, we must first find equivalent rational expressions that do have a common denominator. The least common multiple (LCM) must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any factorizations.
Slide 6- 9 Copyright © 2012 Pearson Education, Inc. Example For each pair of polynomials, find the least common multiple. a) 16a and 24b b) 24x 4 y 4 and 6x 6 y 2 c) x 2 4 and x 2 2x 8 Solution a) 16a = 2 2 2 2 a 24b = 2 2 2 3 b The LCM = 2 2 2 2 a 3 b The LCM is 2 4 3 a b, or 48ab 16a is a factor of the LCM 24b is a factor of the LCM
Slide Copyright © 2012 Pearson Education, Inc. Example continued b) 24x 4 y 4 = 2 2 2 3 x x x x y y y y 6x 6 y 2 = 2 3 x x x x x x y y LCM = 2 2 2 3 x x x x y y y y x x Note that we used the highest power of each factor. The LCM is 24x 6 y 4 c) x 2 4 = (x 2)(x + 2) x 2 2x 8 = (x + 2)(x 4) LCM = (x 2)(x + 2)(x 4) x 2 4 is a factor of the LCM x 2 2x 8 is a factor of the LCM
Slide Copyright © 2012 Pearson Education, Inc. Example For each group of polynomials, find the least common multiple. a) 15x, 30y, 25xyzb) x 2 + 3, x + 2, 7 Solution a) 15x = 3 5 x 30y = 2 3 5 y 25xyz = 5 5 x y z LCM = 2 3 5 5 x y z The LCM is 2 3 5 2 x y z or 150xyz b) Since x 2 + 3, x + 2, and 7 are not factorable, the LCM is their product: 7(x 2 + 3)(x + 2).
Slide Copyright © 2012 Pearson Education, Inc. To Add or Subtract Rational Expressions 1. Determine the least common denominator (LCD) by finding the least common multiple of the denominators. 2. Rewrite each of the original rational expressions, as needed, in an equivalent form that has the LCD. 3. Add or subtract the resulting rational expressions, as indicated. 4. Simplify the result, if possible, and list any restrictions, on the domain of functions.
Slide Copyright © 2012 Pearson Education, Inc. Example Add: Solution 1. First, we find the LCD: 9 = 3 3 12 = 2 2 3 2. Multiply each expression by the appropriate number to get the LCD. LCD = 2 2 3 3 = 36
Slide Copyright © 2012 Pearson Education, Inc. Example continued 3. Next we add the numerators: 4. Since 16x x and 36 have no common factor, cannot be simplified any further. Subtraction is performed in much the same way.
Slide Copyright © 2012 Pearson Education, Inc. Example Subtract: Solution We follow the four steps as shown in the previous example. First, we find the LCD. 9x = 3 3 x 12x 2 = 2 2 3 x x The denominator 9x must be multiplied by 4x to obtain the LCD. The denominator 12x 2 must be multiplied by 3 to obtain the LCD. LCD = 2 2 3 3 x x = 36x 2
Slide Copyright © 2012 Pearson Education, Inc. Example continued Multiply to obtain the LCD and then we subtract and, if possible, simplify. Caution! Do not simplify these rational expressions or you will lose the LCD. This cannot be simplified, so we are done.
Slide Copyright © 2012 Pearson Education, Inc. Example Add: Solution First, we find the LCD: a 2 4 = (a 2)(a + 2) a 2 2a = a(a 2) We multiply by a form of 1 to get the LCD in each expression: LCD = a(a 2)(a + 2).
Slide Copyright © 2012 Pearson Education, Inc. Example continued 3a 2 + 2a + 4 will not factor so we are done.
Slide Copyright © 2012 Pearson Education, Inc. Example Subtract: Solution First, we find the LCD. It is just the product of the denominators: LCD = (x + 4)(x + 6). We multiply by a form of 1 to get the LCD in each expression. Then we subtract and try to simplify. Multiplying out numerators
Slide Copyright © 2012 Pearson Education, Inc. Example continued Removing parentheses and subtracting every term When subtracting a numerator with more than one term, parentheses are important.
Slide Copyright © 2012 Pearson Education, Inc. Example Add: Solution Adding numerators
Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 6.3 Addition and Subtraction of Rational Expressions.
Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions.
Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational.
Chapter 2 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. More on Solving Linear Equations Learn and use the four steps for.
1 MA 1128: Lecture 15 – 6/17/13 Adding and Subtracting Rational Expressions (cont.)
Operations on Rational Expressions Review Test Tomorrow!!!
The Rational Zero Theorem. The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives.
Copyright © 2007 Pearson Education, Inc. Slide R-2 Chapter R: Reference: Basic Algebraic Concepts R.1Review of Exponents and Polynomials R.2Review of.
Copyright©amberpasillas2010. Parts of a Fraction 3 4 = the number of parts = the total number of parts that equal a whole copyright©amberpasillas2010.
Roots & Zeros of Polynomials How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related. 2.5 Zeros of Polynomial Functions.
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.1 Adding and Subtracting Polynomials Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Polynomials By Dr. Julia Arnold Tidewater Community College Copyright 10/19/2002.
FACTORING TRINOMIALS with leading coefficient ax 2 + bx + c to (ax + b)(cx + d)
Factoring a polynomial means expressing it as a product of other polynomials.
Daily Quiz - Simplify the expression, then create your own realistic scenario for the final expression.
Sometimes when looking for the greatest common factor, you may find that it is 1. 18a 5 b a b 6 GCF of 18, 21 and 14 is 1 Not all terms have.
CHAPTER 4 Fraction Notation: Addition, Subtraction, and Mixed Numerals Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 4.1Least Common.
Chapter 14 Rational Expressions Martin-Gay, Developmental Mathematics – Simplifying Rational Expressions 14.2 – Multiplying and Dividing Rational.
1 The Greenebox Factoring Method Copyright 1999 Lynda Greene all rights reserved.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 5.3 Factoring Polynomials.
Combining Like Terms and Distributive Property Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.
Linear Equation in One Variable. A linear equation in one variable is an equation that can be written in the form ax + b = 0 Where a 0 For example: 5x.
Simplify Warm-up. Compare and Contrast Notes - Adding and Subtracting with LIKE Denominators When you are adding or subtracting rational expressions….
6.4 Factoring and Solving Polynomial Equations. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such.
§ 6.6 Rational Equations. Blitzer, Intermediate Algebra, 4e – Slide #87 Solving a Rational EquationEXAMPLE Solve: SOLUTION Notice that the variable x.
Fraction X Adding Unlike Denominators By Monica Yuskaitis.
Section 6.2 Solving Linear Equations Math in Our World.
INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 0 Review of Algebra.
Make sure you know the day and time of the final exam for this section of Math 110: Day: ______ Date:______ Time: ______ to _______ All Math 110 finals.
Surds Surds are a special type of number that you need to understand and do calculations with. The are examples of exact values and are often appear in.
© 2016 SlidePlayer.com Inc. All rights reserved.