Errors - 333202_0A07.qxd A70 2:22 PM Appendix A A.7 Page...

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A70 Appendix A Review of Fundamental Concepts of Algebra What you should learn •A void common algebraic errors. •R e cogn i z e and use algebraic techniques that are common in calculus. Why you should learn it An efficient command of algebra is critical in mastering this course and in the study of calculus. Errors and the Algebra of Calculus A.7 Algebraic Errors to Avoid This section contains five lists of common algebraic errors: errors involving parentheses, errors involving fractions, errors involving exponents, errors involv- ing radicals, and errors involving dividing out. Many of these errors are made because they seem to be the easiest things to do. For instance, the operations of subtraction and division are often believed to be commutative and associative. The following examples illustrate the fact that subtraction and division are neither commutative nor associative. Not commutative Not associative 20 4 s 4 4 2 d Þ s 20 4 4 d 4 2 15 4 5 Þ 5 4 15 8 2 s 6 2 2 d Þ s 8 2 6 d 2 2 4 2 3 Þ 3 2 4 Errors Involving Parentheses Potential Error Correct Form Comment Change all signs when distributing minus sign. Remember the middle term when squaring binomials. occurs twice as a factor. When factoring, apply exponents to all factors. 5 3 2 s x 1 2 d 2 s 3 x 1 6 d 2 5 f 3 s x 1 2 dg 2 s 3 x 1 6 d 2 5 3 s x 1 2 d 2 1 2 5 ab 4 1 1 2 a 21 1 2 b 2 5 1 4 s ab d 1 1 2 a 1 2 b 2 5 1 2 s ab d s a 1 b d 2 5 a 2 1 2 ab 1 b 2 s a 1 b d 2 5 a 2 1 b 2 a 2 s x 2 b d 5 a 2 x 1 b a 2 s x 2 b d 5 a 2 x 2 b Errors Involving Fractions Potential Error Correct Form Comment Leave as Do not add denominators when adding fractions. Multiply by the reciprocal when dividing fractions. Use the property for adding fractions. Use the property for multiplying fractions. Be careful when using a slash to denote division. s 1 y x d 1 2 5 1 x 1 2 5 1 1 2 x x s 1 y x d 1 2 5 1 x 1 2 s 1 y 3 d x 5 1 3 ? x 5 x 3 s 1 y 3 d x 5 1 3 x 1 3 x 5 1 3 ? 1 x 1 3 x 5 1 3 x 1 a 1 1 b 5 b 1 a ab 1 a 1 1 b 5 1 a 1 b 1 x a 2 b 5 1 x a 1 b 2 5 x ab 1 x a 2 b 5 bx a a x 1 b . a x 1 b 5 a x 1 a b Be careful when using a slash to denote division and be sure to find a common denominator before you add fractions.
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Appendix A.7 Errors and the Algebra of Calculus A71 A good way to avoid errors is to work slowly , write neatly , and talk to yourself . Each time you write a step, ask yourself why the step is algebraically legitimate. You can justify the step below because dividing the numerator and denominator by the same nonzero number produces an equivalent fraction . Using the Property for Adding Fractions Describe and correct the error. Solution When adding fractions, use the property for adding fractions: Now try Exercise 17. 1 2 x 1 1 3 x 5 3 x 1 2 x 6 x 2 5 5 x 6 x 2 5 5 6 x 1 a 1 1 b 5 b 1 a ab .
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Errors - 333202_0A07.qxd A70 2:22 PM Appendix A A.7 Page...

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