6.2 Adding and Subtracting Rational Expressions

Transcription

1 8 CHAPTER 6 Rational Epressions Simplify. Assume that no denominator is p - - p q n q n + n + k n k Perform the indicated operation. Write all answers in lowest terms n - # n - n + n + - # 8-6 n + 0. yn + 9 0y 06. yn - 6 y n + 0. yn - y n - y n - # y n - y n - 8 # 6y y n yn + y n + 0 0, yn - + y n, yn + y n + y n + 6. Adding and Subtracting Rational Epressions S Add or Subtract Rational Epressions with a Common Denominator. Identify the Least Common Denominator (LCD) of Two or More Rational Epressions. Add or Subtract Rational Epressions with Unlike Denominators. Adding or Subtracting Rational Epressions with a Common Denominator Rational epressions, like rational numbers, can be added or subtracte We add or subtract rational epressions in the same way that we add or subtract rational numbers (fractions). Adding or Subtracting Rational Epressions with a Common Denominator If P Q and R are rational epressions, then Q P Q + R Q P + R Q and P Q - R Q P - R Q To add or subtract rational epressions with a common denominator, add or subtract the numerators and write the sum or difference over the common denominator. Very Important: Be sure to insert parentheses here so that the entire numerator is subtracte EXAMPLE + b. Add or subtract. z + z y - + y The rational epressions have common denominators, so add or subtract their numerators and place the sum or difference over their common denominator b. z + z + z y - + y - Simplify. Add the numerators and write the result over the common denominator. Subtract the numerators and write the result over the common denominator. Factor the numerator. - + y Subtract the numerators. - - y Use the distributive property. - y Simplify.

2 Section 6. Adding and Subtracting Rational Epressions 9 Add or subtract. 9 z + z b z a - z + a CONCEPT CHECK Find and correct the error. + y y - - y + y - + y - y + y - y + 6 y - Identifying the Least Common Denominator (LCD) of Rational Epressions To add or subtract rational epressions with unlike denominators, first write the rational epressions as equivalent rational epressions with a common denominator. The least common denominator (LCD) is usually the easiest common denominator to work with. The LCD of a list of rational epressions is a polynomial of least degree whose factors include the denominator factors in the list. Use the following steps to find the LCD. Finding the Least Common Denominator (LCD) Step. Step. Factor each denominator completely. The LCD is the product of all unique factors, each raised to a power equal to the greatest number of times that the factor appears in any factored denominator. Answer to Concept Check: + y y - - y + y - + y - y - y - y y - EXAMPLE Find the LCD of the rational epressions in each list. z y, y b. z +, z z - m - m -, m m - 9m -, m - 0m + First we factor each denominator. y # # y y # # y LCD # # # y y -, 6 - b. The denominators z + and z - do not factor further. Thus, LCD z + z - We first factor each denominator. m - m + m - m - 9m - m + m - m - 0m + m - m - LCD m + m + m - The greatest power of is, so we have a factor of. The greatest power of y is, so we have a factor of y.

3 0 CHAPTER 6 Rational Epressions - and - are opposite factors. Notice that - was factored from - so that the factors are identical. Factor each denominator LCD If opposite factors occur, do not use both in the LCD. Instead, factor - from one of the opposite factors so that the factors are then identical. Find the LCD of the rational epression in each list. 6 y, 9 y b. -, + b + b - 6, 8 b - 8b + 6, b b - b - y y - 9, - y Adding or Subtracting Rational Epressions with Unlike Denominators To add or subtract rational epressions with unlike denominators, we write each rational epression as an equivalent rational epression so that their denominators are alike. Adding or Subtracting Rational Epressions with Unlike Denominators Step. Find the LCD of the rational epressions. Step. Step. Step. Write each rational epression as an equivalent rational epression whose denominator is the LCD found in Step. Add or subtract numerators and write the result over the common denominator. Simplify the resulting rational epression. EXAMPLE y + y Perform the indicated operation. b The LCD is y. Write each fraction as an equivalent fraction with denominator y. To do this, we multiply both the numerator and denominator of each fraction by the factors needed to obtain the LCD as denominator. The first fraction is multiplied by so that the new denominator is the LCD. y + y # y # + 6 y + y 6 + y y The second epression already has a denominator of y. Add the numerators.

4 Section 6. Adding and Subtracting Rational Epressions b. The LCD is the product of the two denominators: # - + # # + - # Write equivalent rational epressions. Multiply in the numerators. Add the numerators Simplify the numerator. + - The LCD is either - or -. To get a common denominator of -, we factor - from the denominator of the second rational epression # Write - as - -. Write - - as - # -. Combine the numerators. Simplify. Factor. Simplest form Perform the indicated operation. p q + p q b. y + + y y - z - 8 z z Very Important: Because we are subtracting, notice the sign change on. EXAMPLE Subtract k k - - k + k - k k - - k + k -. k k + k - - k + k - The LCD is k + k - k -. We write equivalent rational epressions with the LCD as denominators. k k + k - - k + k - k # k - k + k - # k - - # k - k + k - # k - k - k k + k - k - - k - k + k - k - k - k - k + k + k - k - k - k + k + k - k - Factor each denominator to find the LCD. Write equivalent rational epressions. Multiply in the numerators. Subtract the numerators. Simplify. Subtract t t - - t - t - 0.

5 CHAPTER 6 Rational Epressions EXAMPLE Add Factor the denominators. The LCD is # # # # Multiply in the numerators. Add the numerators. Simplify. + Add EXAMPLE 6 Perform each indicated operation Factor the denominators. The LCD is - +. # + - # # - + # Add and subtract the numerators. Simplify. Factor the numerator. Divide out common factors. Multiply in the numerators. 6 Perform each indicated operation

6 Section 6. Adding and Subtracting Rational Epressions Graphing Calculator Eplorations A graphing calculator can be used to support the results of operations on rational epressions. For eample, to verify the result of Eample b, graph Y + + Vocabulary, Readiness & Video Check - and Y on the same set of aes. The graphs should be the same. Use a Table feature or a Trace feature to see that this is true. Name the operation(s) below that make each statement true. Addition b. Subtraction Multiplication Division. The denominators must be the same before performing the operation.. To perform this operation, you multiply the first rational epression by the reciprocal of the second rational epression.. Numerator times numerator all over denominator times denominator.. These operations are commutative (order doesn t matter). For the rational epressions y and, perform each operation mentally. y. Addition 6. Subtraction. Multiplication 8. Division Martin-Gay Interactive Videos See Video 6. Watch the section lecture video and answer the following questions. 9. In Eample, why are we told to be especially careful with subtraction when the second numerator has more than one term? 0. In Eample, a - b appears as a factor three times within the two factored denominators. Why does the LCD only contain two factors of (a - b)?. Based on Eample, complete the following statements. To write an equivalent rational epression, you multiply the of the epression by the eact same thing as the denominator. This is the same as multiplying the original rational epression by, which doesn t change the of the original epression. 6. Eercise Set Be careful when subtracting! For eample, Use this eample to help you perform the subtractions y + y - - y - y -. z - z z + z + 6 Add or subtract as indicate Simplify each answer. See Eample.. z z y - y T T or

7 CHAPTER 6 Rational Epressions Find the LCD of the rational epressions in each list. See Eample ,, + +, , y, y, + -, 8 + a + 9, a - a a - b, a - ab + b. a a + 8a + 6, - 9,, - 6. a a + a - 9 -, 0-0, 6 Add or subtract as indicate Simplify each answer. See Eamples a and b y - y Perform the indicated operation. If possible, simplify your answer. See Eample.. 9. a - b + b - a a a Perform each indicated operation. If possible, simplify your answer. See Eamples through y + y - 6y y a a + 0a a a + 6a a + a + + a MIXED Add or subtract as indicate If possible, simplify your answer. See Eamples through y + y y + y a a + a ab a - b + b a + b z z z

8 Section 6. Adding and Subtracting Rational Epressions a + b - a - b. a + b - a - b MIXED (SECTIONS 6., 6.) Perform the indicated operation. If possible, simplify your answer.. a - b # a + b CONCEPT EXTENSIONS Find and correct each error. See the Concept Check in this section Find the perimeter and the area of the square.. a - b, a + b. a a b, a a a + - a + b 6. a +. a b, a b 9. a b # a b - b, +, a + + # a b # a b REVIEW AND PREVIEW + - b 8. a b # a b Use the distributive property to multiply the following. See Section.. 8. a + 6 b 8. a + b 8. a + b 86. y a y - b Find each root. See Section Use the Pythagorean theorem to find the unknown length in each right triangle. See Section feet meters feet meters ft 98. Find the perimeter of the quadrilateral. cm cm cm 8 cm 99. When is the LCD of two rational epressions equal to the product of their denominators? ahint: What is the LCD of and +? b 00. When is the LCD of two rational epressions with different denominators equal to one of the denominators? ahint: What is the LCD of + and + +? b 0. In your own words, eplain how to add rational epressions with different denominators. 0. In your own words, eplain how to multiply rational epressions. 0. In your own words, eplain how to divide rational epressions. 0. In your own words, eplain how to subtract rational epressions with different denominators. Perform each indicated operation. (Hint: First write each epression with positive eponents.) y - + y Use a graphing calculator to support the results of each eercise. 09. Eercise 0. Eercise 8