5.1 Simplifying Algebraic Fractions 5.1 OBJECTIVES 1. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions will be similar to your work in arithmetic. For instance, in algebra, as in arithmetic, many fractions name the same number. You will remember from Chapter 0 that 1 12 2 4 42 8 or 1 13 3 4 43 12 1 2 3 So , , and all name the same number. They are called equivalent fractions. These 4 8 12 examples illustrate what is called the Fundamental Principle of Fractions. In algebra it becomes Rules and Properties: Fundamental Principle of Algebraic Fractions For polynomials P, Q, and R, © 2001 McGraw-Hill Companies PR P Q QR when Q 0 and R 0 This principle allows us to multiply or divide the numerator and denominator of a fraction by the same nonzero polynomial. The result will be an expression that is equivalent to the original one. Our objective in this section is to simplify algebraic fractions by using the fundamental principle. In algebra, as in arithmetic, to write a fraction in simplest form, you divide the numerator and denominator of the fraction by their greatest common factor (GCF). The numerator and denominator of the resulting fraction will have no common factors other than 1, and the fraction is then in simplest form. The following rule summarizes this procedure. Step by Step: NOTE Notice that step 2 uses the Fundamental Principle of Fractions. The GCF is R in the rule above. Step 1 Step 2 To Write Algebraic Fractions in Simplest Form Factor the numerator and denominator. Divide the numerator and denominator by the greatest common factor (GCF). The resulting fraction will be in lowest terms. 395 396 CHAPTER 5 ALGEBRAIC FRACTIONS Example 1 Writing Fractions in Simplest Form (a) Write 18 in simplest form. 30 1 NOTE This is the same as dividing both the numerator 18 and denominator of by 6. 30 1 18 233 2 3 3 3 30 235 2 3 5 5 1 Divide by the GCF. The slash lines indicate that we have divided the numerator and denominator by 2 and by 3. 1 3 4x in simplest form. 6x (b) Write 1 1 4x3 2 2 x x x 2x2 6x 2 3 x 3 1 1 3 2 15x y in simplest form. 20xy4 (c) Write 1 1 1 1 15x3y2 3 5 x x x y y 3x2 20xy4 2 2 5 x y y y y 4y2 1 1 1 1 3a2b in simplest form. 9a3b2 (d) Write 1 1 1 1 3a2b 3 a a b 1 9a3b2 3 3 a a a b b 3ab 1 1 1 1 5 4 (e) Write 10a b in simplest form. 2a2b3 1 1 1 1 1 1 5 2 a a a a a b b b b 5a3b 10a5b4 5a3b 2 3 2a b 2 a a b b b 1 1 1 1 1 1 1 CHECK YOURSELF 1 this chapter build on our factoring work of the last chapter. Write each fraction in simplest form. (a) 30 66 (b) 5x4 15x (c) 12xy4 18x3y2 (d) 5m2n 10m3n3 (e) 12a4b6 2a3b4 In simplifying arithmetic fractions, common factors are generally easy to recognize. With algebraic fractions, the factoring techniques you studied in Chapter 4 will have to be used as the first step in determining those factors. © 2001 McGraw-Hill Companies NOTE Most of the methods of SIMPLIFYING ALGEBRAIC FRACTIONS SECTION 5.1 397 Example 2 Writing Fractions in Simplest Form Write each fraction in simplest form. (a) 2x 4 2(x 2) 2 x 4 (x 2)(x 2) Factor the numerator and denominator. 1 2(x 2) (x 2)(x 2) Divide by the GCF x 2. The slash lines indicate that we have divided by that common factor. 1 2 x2 1 3x2 3 3(x 1)(x 1) (b) 2 x 2x 3 (x 3)(x 1) 1 3(x 1) x3 1 2x2 x 6 (x 2)(2x 3) (c) 2 2x x 3 (x 1)(2x 3) 1 C A U TI O N Pick any value, other than 0, for x and substitute. You will quickly see that x2 2 Z x1 1 x2 x1 Be Careful! The expression x2 is already in simplest form. Students are often tempted x1 to divide as follows: x 2 x 1 is not equal to 2 1 The x’s are terms in the numerator and denominator. They cannot be divided out. Only factors can be divided. The fraction x2 x1 is in its simplest form. © 2001 McGraw-Hill Companies CHECK YOURSELF 2 Write each fraction in simplest form. (a) 5x 15 x2 9 (b) a2 5a 6 3a2 6a (c) 3x2 14x 5 3x2 2x 1 (d) 5p 15 p2 4 398 CHAPTER 5 ALGEBRAIC FRACTIONS Remember the rules for signs in division. The quotient of a positive number and a negative number is always negative. Thus there are three equivalent ways to write such a quotient. For instance, 2 2 2 3 3 3 2 , with the negative 3 sign in the numerator, is the most common way to write the quotient. NOTE The quotient of two positive numbers or two negative numbers is always positive. For example, 2 2 3 3 Example 3 Writing Fractions in Simplest Form Write each fraction in simplest form. 1 quotient is written in the most common way with the minus sign in the numerator. 1 1 1 1 1 5a2b (1) 5 a a b a2 (b) 10b2 (1) 2 5 b b 2b 1 1 1 CHECK YOURSELF 3 Write each fraction in simplest form. 8x3y 4xy2 (a) (b) 16a4b2 12a2b5 It is sometimes necessary to factor out a monomial before simplifying the fraction. Example 4 Writing Fractions in Simplest Form Write each fraction in simplest form. (a) 6x2 2x 3x 1 2x(3x 1) 2x2 12x 2x(x 6) x6 (b) x2 4 x2 (x 2)(x 2) x2 6x 8 (x 2)(x 4) x4 © 2001 McGraw-Hill Companies NOTE In part (a), the final 1 6x2 2 3 x x 2x 2x (a) 3xy (1) 3 x y y y SIMPLIFYING ALGEBRAIC FRACTIONS SECTION 5.1 399 CHECK YOURSELF 4 Simplify each fraction. (a) 3x3 6x2 9x4 3x2 (b) x2 9 x2 12x 27 Reducing certain algebraic fractions will be easier with the following result. First, verify for yourself that 5 8 (8 5) In general, it is true that a b (b a) or, by dividing both sides of the equation by b a, ab (b a) ba ba So dividing by b a on the right, we have NOTE Remember that a and b cannot be divided out because they are not factors. ab 1 ba Let’s look at some applications of that result in Example 5. Example 5 Writing Fractions in Simplest Form Write each fraction in simplest form. (a) 2x 4 2(x 2) 4 x2 (2 x)(2 x) © 2001 McGraw-Hill Companies (b) This is equal to 1. 2(1) 2 2x 2x 9 x2 (3 x)(3 x) 2 x 2x 15 (x 5)(x 3) (3 x)(1) x5 x 3 x5 This is equal to 1. CHAPTER 5 ALGEBRAIC FRACTIONS CHECK YOURSELF 5 Write each fraction in simplest form. (a) 3x 9 9 x2 (b) x2 6x 27 81 x2 CHECK YOURSELF ANSWERS 5 x3 2y2 1 ; (b) ; (c) 2 ; (d) ; (e) 6ab2 11 3 3x 2mn2 5(p 3) 2x2 4a2 (d) 3. (a) ; (b) 3 (p 2)(p 2) y 3b 3 x 3 5. (a) ; (b) x3 x9 1. (a) 5 a3 x5 ; (b) ; (c) ; x3 3a x1 x2 x3 4. (a) 2 ; (b) 3x 1 x9 2. (a) © 2001 McGraw-Hill Companies 400 Name Exercises 5.1 Section Date Write each fraction in simplest form. 1. 16 24 2. ANSWERS 56 64 1. 3. 80 180 4. 18 30 2. 3. 5. 4x5 6x2 6. 10x2 15x4 4. 5. 7. 9x3 27x6 8. 25w6 20w2 6. 7. 10a2b5 9. 25ab2 18x4y3 10. 24x2y3 8. 9. 10. 42x3y 11. 14xy3 18pq 12. 45p2q2 11. 12. 2 13. 2xyw 6x2y3w3 2 2 14. 3c d 6bc3d 3 13. 14. 5 5 15. 10x y 2x3y4 6 3 16. 3bc d bc3d 15. © 2001 McGraw-Hill Companies 16. 17. 4m3n 6mn2 18. 15x3y3 20xy4 17. 18. 19. 8ab3 16a3b 20. 14x2y 21xy4 19. 20. 401 ANSWERS 21. 21. 8r2s3t 16rs4t3 22. 10a3b2c3 15ab4c 23. 3x 18 5x 30 24. 4x 28 5x 35 25. 3x 6 5x 15 26. x2 25 3x 15 27. 6a 24 a2 16 28. 5x 5 x2 4 29. x2 3x 2 5x 10 30. 4w2 20w w 2w 15 31. x2 6x 16 x2 64 32. y2 25 y2 y 20 33. 2m2 3m 5 2m2 11m 15 34. 6x2 x 2 3x2 5x 2 35. p2 2pq 15q2 p2 25q2 36. 4r2 25s2 2r 3rs 20s2 37. 2x 10 25 x2 38. 3a 12 16 a2 39. 25 a2 a2 a 30 40. 2x2 7x 3 9 x2 41. x2 xy 6y2 4y2 x2 42. 16z2 w2 2w 5wz 12z2 22. 23. 24. 25. 26. 27. 28. 29. 30. 2 31. 32. 33. 34. 35. 36. 2 38. 39. 40. 41. 42. 402 2 © 2001 McGraw-Hill Companies 37. ANSWERS x2 4x 4 43. x2 43. 4x2 12x 9 44. 2x 3 44. 45. xy 2y 4x 8 2y 6 xy 3x 46. ab 3a 5b 15 15 3a2 5b a2b 47. y7 7y 48. 5y y5 45. 46. 47. 49. The area of the rectangle is represented by 6x2 19x 10. What is the length? 48. 49. 50. 3x 2 51. 50. The volume of the box is represented by (x2 5x 6)(x 5). Find the polynomial 52. that represents the area of the bottom of the box. 53. x2 54. 51. To work with algebraic fractions correctly, it is important to understand the difference between a factor and a term of an expression. In your own words, write difinitions for both, explaining the difference between the two. 55. 52. Give some examples of terms and factors in algebraic fractions, and explain how both are affected when a fraction is reduced. 53. Show how the following algebraic fraction can be reduced: © 2001 McGraw-Hill Companies x2 9 4x 12 Note that your reduced fraction is equivalent to the given fraction. Are there other algebraic fractions equivalent to this one? Write another algebraic fraction that you think is equivalent to this one. Exchange papers with another student. Do you agree that their fraction is equivalent to yours? Why or why not? 54. Explain the reasoning involved in each step of reducing the fraction 55. Describe why 42 . 56 3 27 and are equivalent fractions. 5 45 403 ANSWERS a. Getting Ready for Section 5.2 [Section 0.2] b. Perform the indicated operations. 3 10 5 (c) 12 7 (e) 20 11 (g) 6 c. (a) d. e. f. 4 10 1 12 9 20 2 6 5 4 8 8 7 3 (d) 16 16 13 5 (f) 8 8 5 7 (h) 9 9 (b) g. Answers h. 1. 2 3 1 2ab3 3x2 1 9. 11. 2 13. 3 3x 5 y 3xy2w 2m2 b2 r 3 3(x 2) 17. 19. 21. 23. 25. 2 2 3n 2a 2st 5 5(x 3) x1 x2 m1 p 3q 29. 31. 33. 35. 5 x8 m3 p 5q a 5 x 3y (y 4) 39. 41. 43. x 2 45. a6 2y x y3 4 9 3. 15. 5x2y 6 a4 2 37. x5 27. 47. 1 7 10 2x3 3 49. 2x 5 b. 1 8 c. 7. 51. 1 3 d. 53. 5 8 e. 4 5 55. f. 1 g. 3 2 h. 4 3 © 2001 McGraw-Hill Companies a. 5. 404

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