have in common. The greatest common factor of a polynomial should be factored out first before any further factoring is completed. Example: 3r6+27r4+15r2=3r2(r4+9r2+5) When multiplying variables‚ add the exponents. r^2•r^4=rr•rrrr=r6 When factoring a GCF‚ subtract the exponents. To factor r^2 from r^6: r^6−2=r^4 rrrrrr=(rr)(rrrr)=r2(r4) Difference of Squares Binomials A difference of squares binomial includes a perfect square term subtracted by another perfect square term. Pattern: a^2−b^2=(a+b)(a−b)
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MATH INVESTIGATION 4.2 FACTORIZATIONS on the Math Investigator determines if a number is prime or composite. If a number is composite‚ it prints all its factors‚ the number of factors‚ and its prime factorization. The numbers 1‚ 2‚ 4‚ and 6 have 1‚ 2‚ 3‚ and 4 factors‚ respectively: 1 has only 1 as a factor; 2 has 1 and 2 as factors; 4 has 1‚ 2‚ and 4 as factors; and 6 has 1‚ 2‚ 3‚ and 6 as factors. These factors are illustrated by the rectangles shown here. Starting Points for Investigations
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squares (1 x 1) that a diagonal passes through. The rectangles are a) 2 by 7 and b) 3 by 4. In order to come up with an appropriate solution‚ one must consider the greatest common divider and its possibilities. The numbers need to be considered‚ are they prime or composite numbers? Is the slope a factor in finding the answer? The first answer provided was to find out the number of squares the diagonal line goes through‚ first divide the L (length) of the rectangle by the W (width). This will not be a whole
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Lesson Guides in Elementary Mathematics Grade V Copyright © 2003 All rights reserved. No part of these lesson guides shall be reproduced in any form without a written permission from the Bureau of Elementary Education‚ Department of Education. The Mathematics Writing Committee GRADE 5 Region 3 Nelia D. Bamba – Pampanga Magdalena P. Garcia – Angeles City Region 4 Gundalina C. Gonzales – Batangas Gerlie Ilagan – Quezon Consuelo Caringal – Batangas National Capital Region (NCR) Emma S. Makasiray –
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(pattern) 10 = 3.3333... 3 5 = 0.8333... 6 0.1010010000100001... 3 = 1.7320508075... log10 5 = 0.698970004336... 3 = 1.37050... • Real Numbers can be represented geometrically as points on a number line called Real Line Example Prime Numbers - any natural number‚ greater than 1‚ only divisible by itself and 1 Integers (Z) – any positive and negative natural number including ‘0’ Zero – the number represents ‘none’ ‘nothing’ Even Numbers – any number divisible by
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mainly to compute the HCF of two positive integers. The Fundamental Theorem of Arithmetic‚ on the other hand‚ has to do something with multiplication of positive integers. You already know that every composite number can be expressed as a product of primes in a unique way — this important fact is the Fundamental Theorem of Arithmetic. Again‚ while it is a result that is easy to state and understand‚ it has some very deep and significant applications in the field of mathematics. We use the Fundamental
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in the table to answer the following questions. If necessary‚ check fractions other than those in the table. a. Which prime numbers appear in the prime factorization of the denominators of those fractions whose decimal equivalents terminate? 2. b. Does the prime factorization of the denominator of any of the fractions whose decimal equivalents repeat contain only the primes in part a? If so‚ which fractions? c. How can you predict whether the decimal equivalent of a given fraction terminates
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Multiply the remaining power of 10 5.6 54. A student simplifies the expression 4 x3 - 1 4 x2 to x - 1. Explain the student’s error. The answer should be 1 6.2 Factor the trinomial completely. 66. -y3 + 9y2 - 14y The GCF for this problem is -1y -y(y^2-9y+14)
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Annotated Instructor’s Edition Elementary and Intermediate Algebra Third Edition George Woodbury College of the Sequoias Addison-Wesley Boston Amsterdam Delhi Columbus Indianapolis London New York Madrid San Francisco Milan Munich Seoul Upper Saddle River Paris Montreal Taipei Toronto Tokyo Cape Town Mexico City Dubai Sao Paulo Sydney Hong Kong Singapore Editorial Director: Christine Hoag Editor in Chief: Maureen O’Connor Acquisitions Editor: Dawn Giovaniello Executive Project
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by Complete Induction? Archive for History of the Exact Sciences 55 (2000)‚ 57–76. [Li33] F.A Lindemann‚ The Unique Factorization of a Positive Integer. Quart. J. Math. 4‚ 319– 320‚ 1933. [Mu63] A.A. Mullin‚ Recursive function theory (A modern look at a Euclidean idea). Bulletin of the American Mathematical Society 69 (1963)‚ 737. [Ro63] K. Rogers‚ Classroom Notes: Unique Factorization. Amer. Math. Monthly 70 (1963)‚ no. [Ze34] E. Zermelo‚ Elementare Betrachtungen zur Theorie der Primzahlen. Nachr
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