* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chapter 1 Linear Equations and Graphs
Numbers (TV series) wikipedia , lookup
Positional notation wikipedia , lookup
History of mathematical notation wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Mathematics and architecture wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Mathematics and art wikipedia , lookup
Philosophy of mathematics wikipedia , lookup
Factorization wikipedia , lookup
Mathematics wikipedia , lookup
History of mathematics wikipedia , lookup
Critical mathematics pedagogy wikipedia , lookup
Secondary School Mathematics Curriculum Improvement Study wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Ethnomathematics wikipedia , lookup
Appendix A Basic Algebra Review Section A-4 Operations on Rational Expressions Operations on Rational Expressions Reducing to Lowest Terms Multiplication and Division Addition and Subtraction Compound Fractions Barnett/Ziegler/Byleen College Mathematics 12e 2 Rational Expressions A quotient of two algebraic expressions (division by 0 excluded) is called a fractional expression. If both the numerator and the denominator are polynomials, the fractional expression is called a rational expression. 1 x 3 2x 5 x x7 3x 2 5x 1 Barnett/Ziegler/Byleen College Mathematics 12e x 2 2x 4 1 3 Rational Expressions AGREEMENT Variable Restriction Even though not always explicitly stated, we always assume that variables are restricted so that division by 0 is excluded. For example, given the rational expression 2x 5 x x 2 x 3 the variable x is understood to be restricted from being 0, –2, or 3, since these values would cause the denominator to be 0. Barnett/Ziegler/Byleen College Mathematics 12e 4 Reducing to Lowest Terms Fundamental Property of Fractions If a, b, and k are real numbers with b, k 0, then For example, given the rational expression ka a kb b the variable x is understood to be restricted from being 0, –2, or 3, since these values would cause the denominator to be 0. Barnett/Ziegler/Byleen College Mathematics 12e 5 Reducing to Lowest Terms Using this property from left to right to eliminate all common factors from the numerator and the denominator of a given fraction is referred to as reducing a fraction to lowest terms. Using the property from right to left–that is, multiplying the numerator and denominator by the same nonzero factor–is referred to as raising a fraction to higher terms. Reduce to lowest terms. x x 2 x 2 2x 4 x 4 8x 3 2 3x 2x 8x x x 2 3x 4 Barnett/Ziegler/Byleen College Mathematics 12e x 2 2x 4 3x 4 6 Multiplication and Division Multiplication and Division If a, b, c, and d are real numbers, then a c ac 1. , b, d 0 b d bd a c a d 2. , b, c, d 0 b d b c Barnett/Ziegler/Byleen College Mathematics 12e 7 Multiplication and Division Perform the indicated operation and reduce to lowest terms. 10x 3 y 4 x 2 12x 10x 3 y x2 9 2 2 3xy 9y x 9 3xy 9y 4 x 12x 10x 3 y x 3x 3 3y x 3 4x x 3 5 x2 3 11 31 21 x 3 x 3 3 y x 3 4 x x 3 10x y 5x 2 6 Barnett/Ziegler/Byleen College Mathematics 12e 8 Addition and Subtraction Addition and Subtraction For a, b, and c real numbers, a c ac 1. , b b b a c ac 2. , b b b Barnett/Ziegler/Byleen College Mathematics 12e b0 b0 9 Addition and Subtraction Least Common Denominator The least common denominator (LCD) of two or more rational expressions is found as follows: 1. Factor each denominator completely, including integer factors. 2. Identify each different factor from all the denominators. 3. Form a product using each different factor to the highest power that occurs in any one denominator. This product is the LCD. Barnett/Ziegler/Byleen College Mathematics 12e 10 Addition and Subtraction Combine into a single fraction and reduce to lowest terms. 1 1 2 2 x 1 x x 1 LCD = x(x – 1)(x + 1) 1 1 2 x 1 x x 1x 1 x x 1 x 1x 1 2x x x 1x 1 x 2 x x 2 1 2x 1 x x x 1x 1 x x 1x 1 x 1 x x 1 x 1 Barnett/Ziegler/Byleen College Mathematics 12e 1 x x 1 11 Compound Fractions A fractional expression with fractions in its numerator, denominator, or both is called a compound fraction. It is often necessary to represent a compound fraction as a simple fraction–that is (in all cases we will consider), as the quotient of two polynomials. We will use the two different methods. Barnett/Ziegler/Byleen College Mathematics 12e 12 Compound Fractions Express as a simple fraction reduced to lowest terms. Use division of rational forms. 1 1 5h 5 1 1 h 5 h 5 1 h 55h 1 5 5 h h h 5 5 h h 1 5 5 h Barnett/Ziegler/Byleen College Mathematics 12e 13 Compound Fractions Express as a simple fraction reduced to lowest terms. Multiply the numerator and denominator by the LCD of all fractions. y 2 x y x x 2 2 y x y 2 2 x y x 2 2 y x y y x x 2 2 y 2 2 x y x y 2 2 y x x 2 2 y x y x 2 y2 x y y x 3 3 xy x y 3 3 y 2 xy x 2 xy y x Barnett/Ziegler/Byleen College Mathematics 12e x y2 x y y x y 2 xy x 2 xy y x y x or x 2 xy y 2 xy x y 14