R 1 R T. Multiply and Divide Rational Expressions. Simplify each expression and state any restrictions on the variables. _. _ 4x2

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1 . R 1 object f R Skills You Need: Operations With Rational Expressions R R T R 1 R R 3 f d o d i d o d i image The ability to manipulate rational expressions is an important skill for engineers, scientists, and mathematicians. Some examples of such situations are the calculation of the resistance in parallel circuits and the calculation of the focal length in curved lenses. Example 1 Multiply and Divide Rational Expressions Simplify each expression and state any restrictions on the variables. a) 4x 3x 1x3 x b) 10ab 15a 4a 1b a) Method 1: Multiply and Then Simplify 4x 3x 1x3 x 5 48x5 6x 5 8x 3 Thus, 4x 3x 1x3 x 5 8x3, x 0. Method : Simplify and Then Multiply 4x 3x 1x3 x x 3x 1x3 x, x 0 5 x 4x 5 8x 3 Thus, 4x 3x 1x3 x 5 8x3, x 0. Multiply the numerators and multiply the denominators. 5 48x5 6x, x 0 Divide by the common factors. 1 4 Divide by the common factors. 88 MHR Functions 11 Chapter

2 10ab 15a 4a 1b 5 10ab 1b 4a 15a 10ab4 60a 3 10ab4 60a, a 0 3 b4 a In the original expression, both a and b were in the denominator, so neither of them can be equal to zero. So, 10ab 15a 4a 1b 5 b4, a 0, b 0. a Multiply by the reciprocal. Multiply the numerators and multiply the denominators. Divide by the common factors. Example Multiply and Divide Rational Expressions Involving Polynomials Simplify and state any restrictions. a) a a 0a 3a 5a 10a x 8x x 3x 10 4x x 9x 0 a) a a 0a 3a 5a 10a a(a ) 0a 3a 5a(a ) 4 1 Factor binomials where possible. a(a ) 0a, a, a 0 Divide by the common factors. 3a 5a(a ) 1 4a Multiply the numerators and multiply 3 4a the denominators. 3 So, a a 0a 3a 5a 10a 4a, a, a Skills You Need: Operations With Rational Expressions MHR 89

3 x 8x x 3x 10 4x x 9x 0 x(x 4) (x 5)(x ) 4x (x 4)(x 5) x(x 4) (x 5)(x ) (x 4)(x 5) 4x x(x 4) (x 5)(x ) (x 4)(x 5), x, x 0, x 5 1 4x (x 4) x(x ) When considering restrictions, you must include any instance where the denominator can be zero. From the original expression, this occurs when x 5 5 0, x 5 0, and x When the second rational expression is inverted, then its denominator can be zero when x 5 0. So, x 8x x 3x 10 4x (x 4) x 9x 0 x(x ), x, x 0, x 4, x 5. Factor binomials and trinomials where possible. Multiply by the reciprocal. Divide by any common factors. Example 3 Add and Subtract Rational Expressions With Monomial Denominators Simplify and state the restrictions. a) 1 1 5x x b) ab ab b b a) Start by determining the least common multiple (LCM) of the denominators. 5x 5 (5)(x) x 5 ()(x) (5)()(x) 5 10x The LCM is the least common denominator (LCD) of the two rational expressions. 1 5x 1 x Thus, 1 5x 1() 5x() 10x 7 10x 1 x 1(5) x(5) 5 10x 7, x 0. 10x Multiply each rational expression by a fraction equal to 1 that makes each denominator 10x. Add the numerators. 90 MHR Functions 11 Chapter

4 b) Determine the LCM of the denominators. ab 5 ()(a)(b)(b) b 5 ()(b) ()(a)(b)(b) 5 ab The LCD is ab. ab b ab b 5 ab ab (b )(ab) Multiply each rational expression by a fraction equal to 1 that b(ab) makes each denominator ab ab ab. ab ab ab ab ab (1 ab) ab 1 ab ab Thus, ab ab b 1 ab, a 0, b 0. b ab Subtract the numerators. Factor from the numerator. Divide by the common factor of. Example 4 Add and Subtract Rational Expressions With Polynomial Denominators Simplify and state the restrictions. a) x x 7 x 3 x x 9 x x 48 x 9 x x 30 a) There are no common factors in the denominators, so the LCD is just (x 3)(x ). x x 7 x 3 x (x 5)(x ) (x 7)(x 3) (x 3)(x ) (x )(x 3) x 7x 10 (x 3)(x ) 10x 1 x (x 3)(x ) x 3x 31 (x 3)(x ) Thus, x 5 x 3 x 7 Multiply each rational expression by a fraction equal to 1 that makes each denominator (x 3)(x + ). Add the numerators. x 5 3x 31 x, x, x 3. (x 3)(x ). Skills You Need: Operations With Rational Expressions MHR 91

5 b) Determine the LCM of the denominators. x x 48 5 (x 8)(x 6) x x 30 5 (x 6)(x 5) (x 8)(x 6)(x 5) The LCD is (x 8)(x 6)(x 5). x 9 x x 48 x 9 x x 30 (x 9)(x 5) (x 8)(x 6)(x 5) (x 9)(x 8) (x 6)(x 5)(x 8) x 14x 45 (x 8)(x 6)(x 5) x x 7 (x 8)(x 6)(x 5) 15x 117 (x 8)(x 6)(x 5) Thus, x 9 x x 48 x 9 x x 30 15x 117 (x 8)(x 6)(x 5), x 8, x 5, x 6. Multiply each rational expression by a fraction equal to 1 that makes each denominator (x + 8)(x 6)(x + 5). Add the numerators. Example 5 Bicycle Relay Raj and Mack are competing as a relay team in a 50-km cycling race. There are two legs in the race. Leg A is 30 km and leg B is 0 km. a) Assuming that each cyclist travels at a different average speed, determine a simplified expression to represent the total time of the race. b) If Raj can maintain an average speed of 35 km/h and Mack an average speed of 5 km/h, determine the minimum time it will take to complete the race. a) For any distance-speed-time calculation, the expression for the time, t, is given by t d v, where d represents the distance and v represents the speed. To calculate the total time, add the times for the two legs. Let t A and t B represent the times and v A and v B represent the speeds of legs A and B, respectively. 9 MHR Functions 11 Chapter

6 t 5 ta tb 30 0 v v A B 30v 0v Write with a common denominator. 30v 0v v v Add the numerators. B A v v v v A B A B B A A B b) It makes sense that for the minimum time, the fastest person should ride the longest leg. So, Raj will ride leg A and Mack will ride leg B. 30(5) 0(35) 35(5) t 1.66 Substitute the value for each person s speed. It will take the team approximately 1.66 h to complete the race. Key Concepts When multiplying or dividing rational expressions, follow these steps: Factor any polynomials, if possible. When dividing by a rational expression, multiply by the reciprocal of the rational expression. Divide by any common factors. Determine any restrictions. When adding or subtracting rational expressions, follow these steps: Factor the denominators. Determine the least common multiple of the denominators. Rewrite the expressions with a common denominator. Add or subtract the numerators. Simplify and state the restrictions. Communicate Your Understanding (x 3)(x 6) (x 6)(x 8). What are the restrictions (x 4)(x 5) (x 4)(x 7) C1 Describe how you would simplify on the variable? x 5, x 4, x 1, x. C Write two rational expressions whose product is x x 3 x 3 x. C3 A student simplifies the expression and gets an answer of What did the student probably do incorrectly to get this answer? 5 7x. What are the restrictions on the variable? C4 Describe how you would simplify x 3 x 1. Skills You Need: Operations With Rational Expressions MHR 93 Functions 11 CH0.indd 93 6/10/09 4:01:53 PM

7 A Practise For help with questions 1 and, refer to Example Simplify and state the restrictions on the variables. a) 14y 11y 11x 7x b) 0x3 35x5 7x 4x c) 15b3 0b 4b 30b d) 30ab18a 1a 45b. Simplify and state the restrictions on the variables. a) 5x 5x 9y 18y 55xy 8y 1 48x c) 6ab 4a b 3 39a4 1b d) b 4 3a 16ab 6c 4c 3 For help with questions 3 to 6, refer to Example. 3. Simplify and state the restrictions on the variable. a) 5 x 10 x 10 x 1 5 x x x 1 c) x 5 x 3 x 3 d) x 3 x 8 x 7 x 8 x 3 4. Simplify and state the restrictions on the variable. a) 3x 4x 6 1x 18x 3x 30 4x 4 x 8x 1x 3x 18 c) x 10x 1 x x 3 x 9x 14 d) x x 15 x 6 x 9x 18 x 5. Simplify and state the restrictions on the variable. a) x x 1 x 1 x x x 3 1 x 3 c) x 1 x 10 x 1 d) x 7 x 7 x 5 x 3 x 3 6. Simplify and state the restrictions on the variable. a) x 15x 4x 4 3x 3x 18 6x 8x 7 9x x 18 c) x 15x 6 6x 3x 10 x 30x 3 d) x 11x 4 x 8 x x 3 x 1 For help with question 7, refer to Example Simplify and state any restrictions. a) x 1 x 1 x 10 x c) 1 3x 4x d) 7 3 6x 8x e) 3 5 ab 4b f) a b 4b g) a 4 a a b 3ab h) 4 ab ab 9ab 6a b For help with questions 8 and 9, refer to Example Simplify and state the restrictions. a) 1 x 6 1 x 6 1 x 8 3 x 9 c) x 10 x 3 x 6 x 4 d) x x x 1 x 9. Simplify and state the restrictions. a) x x 9x 8 x 8 x 3 x x x 3x 10 c) x x 3x 3x x 8x 7 d) x 4 x 11 x 1 x 8x MHR Functions 11 Chapter

8 B Connect and Apply For help with question 10, refer to Example Alice is in a 0-km running race. She always runs the first half at an average speed of km/h faster than the second half. a) Let x represent her speed in the first half. Determine a simplified expression in terms of x for the total time needed for the race. b) If Alice runs the first half at 10 km/h, how long will it take her to run the race? 11. Binomial expressions can differ by a factor of 1. Factor 1 from one of the denominators to identify the common denominator. Then, simplify each expression and state the restrictions. a) 1 x 1 x x 7 x 9 x 3 3 x c) a 1 a 4 d) b 3 b 6 5 a a 5 4b 1 1 4b 1. An open-topped box is to be created from a 100-cm by 80-cm piece of cardboard by cutting out a square of side length x from each corner. a) Express the volume of the box as a function of x. Representing Connecting Reasoning and Proving Problem Solving Communicating 100 cm b) Express the surface area of the open-topped box as a function of x. Selecting Tools Reflecting x x c) Write a simplified expression for the ratio of the volume of the box to its surface area. d) Based on your answer in part c), what are the restrictions on x? What are the restrictions in the context? 80 cm 13. Resistors are components found on most circuit boards and in most electronic devices. Since resistors do not come in every size, they have to be arranged in various ways to get the needed resistance. When three resistors are in parallel, then the total resistance, R T, can be calculated using the equation , R T R 1 R R 3 where each of the resistances is in ohms (Ω). a) Determine an expression for the total resistance, R T. b) Determine an expression for the total resistance if R 1 5 R 5 R 3. c) Determine an expression for the total resistance if R 1 5 R 5 6R Consider a cylinder of height h and radius r. a) Determine the ratio of the volume of the cylinder to its surface area. b) What restrictions are there on r and h? 15. Olivia can swim at an average rate of v metres per second in still water. She Reasoning and Proving Representing Problem Solving has two races Connecting Reflecting coming up, one in a lake with no current Communicating and the other in a river with a current of 0.5 m/s. Each race is 800 m, but in the river race she swims the first half against the current and the second half with the current. a) Determine an expression for the time for Olivia to complete the lake swim. b) Determine an expression for the time for Olivia to complete the river swim. c) Olivia thinks that if she swims each race exactly the same and the current either slows her down or speeds her up by 0.5 m/s, both races will take the same amount of time. Is she correct? Explain. r h Selecting Tools. Skills You Need: Operations With Rational Expressions MHR 95

9 16. Use Technology a) Use graphing technology to graph 1 f (x) x 1 x. b) Rewrite the function using a common denominator. Then, graph the rewritten function. c) Compare the graphs. Identify how the restrictions affect the graph. 19. Simplify the expression and state any restrictions. x 8 x 9x 10 13x 40 x x x 1 x 10x 16 x 9 0. a) Evaluate the expression Achievement Check 17. a) Simplify the expressions for A and B, where A x 4 x 9x 0 and B 3x 9x. State the x 3x 18 restrictions. b) Are the two expressions equivalent? Justify your answer. c) Write another expression that appears to be equivalent to each expression in part a). d) Determine A B, AB, and B A. C Extend 18. Archimedes of Syracuse (87 1 bce) studied many things. One was the relationship between a cylinder and a sphere. In particular, he looked at the situation where the sphere just fits inside the cylinder so that they have the same radius and the height of the cylinder equals the diameter of the sphere. a) Determine the ratio of the volume of the sphere to the volume of the cylinder in this situation. b) Determine the ratio of the surface area of the sphere to the surface area of the cylinder in this situation. c) What seems to be true about your answers from parts a) and b)? Connections Archimedes was so fond of the sphere and cylinder relationship that he had the image of a sphere inscribed in a cylinder engraved on his tombstone. b) On a scientific calculator, locate the e x button and enter e 1. Compare your answer for part a) to the constant e. c) The pattern shown in part a) continues on forever. What are the next three steps in this pattern? How do they affect your comparison from part b)? 1. Math Contest When n is divided by 4, the remainder is 3. When 6n is divided by 4, the remainder is A 1 B C 3 D 0. Math Contest The sum of the roots of (x 4x 3)(x 3x 10) (8x 8x 16) 5 0 is A 7 B 6 C 6 D 8 3. Math Contest Given f (x) 36 35, what is the x x 1 smallest integral value of x that gives an integral value of f (x)? 4. Math Contest Given x 3y 4z 5 5, then x 3 y 4 z 5 x y z is A 40 B 40 C 00 D MHR Functions 11 Chapter