1.6 Multiplying and Dividing Rational Expressions

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1 1.6 Multiplying and Dividing Rational Epressions The game of badminton originated in England around Badminton is named after the Duke of Beaufort s home, Badminton House, where the game was first played. The International Badminton Federation now has over 50 member countries, including Canada. INVESTIGATE & INQUIRE In a doubles game of badminton, there are four service courts. The width of each service court is half the width of the whole court. The length of each service court is one third of the distance between the long service lines. The width of the whole court and the distance between the long service lines can be modelled by the epressions shown. left service court right service court 1. a) Write an epression that represents the width of each service court. b) Write an epression that represents the length of each service court. c) Write an epression that represents the area of each service court. Leave your answer in the form, where each numerator is a monomial and and represent whole numbers. MHR Chapter 1 net doubles side line right service court left service court doubles long service line

2 . a) Use the dimensions and to write an epression that represents the whole area shown. b) Simplify the epression.. a) What fraction of the whole area does each service court cover? b) Write this fraction of the epression you wrote in question b). Do not simplify.. How is the epression you wrote in question b) related to your epression for the area of each service court in question 1c)? Eplain. 5. A large rectangle of width 1 and length + 5 is divided into 1 small rectangles, as shown. a) Write an epression that represents the width of each small rectangle. b) Write an epression that represents the length of each small rectangle. c) Write an epression that represents +5 the area of each small rectangle. Leave your answer in the form, where each numerator is a 1 binomial, and and represent whole numbers. 6. a) Use the dimensions 1 and + 5 to write an epression that represents the area of the large rectangle. b) Epand and simplify the epression. 7. a) What fraction of the large rectangle does each small rectangle cover? b) Write this fraction of the epression you wrote in question 6b). 8. How are the epressions from questions 5c) and 7b) related? Eplain. 9. Write a rule for multiplying rational epressions. 10. Multiply. Simplify the product, if possible. a) y a b) c) a b d) t e) f) t + 1 t + y 9y In the epressions that model the badminton court, represents about 6 m. Find the area of each service court, in square metres. y 1.6 Multiplying and Dividing Rational Epressions MHR 5

3 Rational epressions can be multiplied in the same way that fractions are multiplied. Multiply the numerators: Multiply the denominators: Divide by the common factor: = P R P R PR For rational epressions and, =, Q, S 0. Q S Q S QS = = EXAMPLE 1 Multiplying Rational Epressions a 10b Simplify. State the restrictions on the variables. b 9a SOLUTION Multiply the numerators: Multiply the denominators: 5ab Divide by the common factors: = Eclude values for which b = 0 or 9a = 0. b = 0 a = 0 So, a 0, b 0. a 10b 5ab Therefore, =, a 0, b 0. b 9a = a b 0a b 18a b 10b 9a 6 MHR Chapter 1

4 When multiplying some rational epressions, you may find it easier to factor the numerators and the denominators first. EXAMPLE Multiplying Rational Epressions Involving Polynomials + 6 Simplify. State the restrictions on the variable SOLUTION + 15 ( + )( ) Factor: = ( + 5)( ) Multiply the numerators: ( + )( )( ) = Multiply the denominators: ( + 5)( )( ) 1 1 ( + )( ) ( ) Divide by the common factors: = ( + 5)( ) ( ) 1 1 = Eclude values for which ( + 5)( ) = 0 or = = 0 or = 0 or = 0 = 5 = = Therefore, =,,, Rational epressions can be divided in the same way fractions are divided Multiply by the reciprocal: = 5 1 = 15 P R P R P S PS For rational epressions and, = =, Q, R, S 0. Q S Q S Q R QR Note the restrictions on Q, R, and S. 1.6 Multiplying and Dividing Rational Epressions MHR 7

5 EXAMPLE Dividing Rational Epressions ab 1a b Simplify. State the restrictions on the variables. 5c 15c SOLUTION ab 5c 1a b 15c ab Multiply by the reciprocal: = 5c Multiply the numerators: 0abc = Multiply the denominators: 70a b c c Divide by the common factors: = 7ab 15c 1a b Eclude values for which 5c = 0, 1a b = 0, or 15c = 0. 5c = 0 when c = 0. 1a b = 0 when a = 0 or b = 0. 15c = 0 when c = 0. So, a 0, b 0, c 0. ab 1a b c Therefore, =, a 0, b 0, c 0. 5c 15c 7ab When dividing some rational epressions, you may find it easier to factor the numerators and the denominators first. EXAMPLE Dividing Rational Epressions Involving Polynomials Simplify. State the restrictions on the variable SOLUTION ( 5)( + ) ( + )( + 5) Factor: = ( 6) ( 6)( 6) 8 MHR Chapter

6 ( 5)( + ) ( 6)( 6) Multiply by the reciprocal: = ( 6) ( + )( + 5) Multiply the numerators: ( 5)( + )( 6)( 6) = Multiply the denominators: ( 6)( + )( + 5) 1 1 ( 5)( + ) ( 6) ( 6) Divide by the common factors: = ( 6) ( + ) ( + 5) 1 1 ( 5)( 6) = ( + 5) Eclude values for which ( 6) = 0, ( + )( + 5) = 0, or ( 6)( 6) = 0. = 0 or 6 = 0 + = 0 or + 5 = 0 6 = 0 = 6 = = 5 = ( 5)( 6) Therefore, =, 0, 6,, ( + 5) Key Concepts P R P R PR For rational epressions and, =, Q, S 0. Q S Q S QS To multiply rational epressions, a) factor any binomials and trinomials b) multiply the numerators and multiply the denominators c) divide by common factors d) determine and eclude the values of the variable that make the denominators 0 P R P R P S PS For rational epressions and, = =, Q, R, S 0. Q S Q S Q R QR To divide rational epressions, a) factor any binomials and trinomials b) multiply by the reciprocal of the divisor c) multiply the numerators and multiply the denominators d) divide by common factors e) determine and eclude the values of the variable that make the denominators Multiplying and Dividing Rational Epressions MHR 9

7 Communicate Your Understanding 1a 1. Write two rational epressions whose quotient is. 5b + 9. Describe how you would simplify a) Describe how you would simplify. b) What are the restrictions on the variable? Practise In each of the following, state any restrictions on the variables. A 1. Simplify. y 8 7 a) b) y 1 5n m c) d) 6 15n Simplify. 1 y y a) b) t 6t c) d) m m r e) f) r 5 8. Simplify. 8y 8m 5m a) b) y 9 n 6n 1y 1 a 8a c) d) t 7 7b y 7 1m 8m 15a b e) f) 8abc 5t 15 c. Simplify. 16ab 5 y 6 y 9y a) b) 9 10mn 8a 5mn y b 5y 10y c) d) 1a ab b 6 ab y 9 y e) 6 y y a b c f) 6c ab 5. Simplify. m + a) b) 6 5 5(y ) y + 1 ( + 1) c) d) y a b 8ab (m + ) 6m e) f) (a + b) a + b 5m (m + ) 6. Simplify m a) b) m + a + 6 a + c) d) 9a a + 7y y + 1 m 5 e) f) 1y y 9 m 16 y m m m 10 m MHR Chapter 1

8 6 y g) h) 8 y Simplify a) a + 7a + 1 a a 6 b) a + a + a 9 m m m 7m + 1 c) m + 5m m + m 15 1a 19a + 5 a d) a 9 a e) w 5w 1w + w 6 f) 8w + w 1 8w w Simplify. y 0y + y 8y a) 8y + 15y y 6y + y 10y + 1y b) y y 9y a + 15ab + 56b a + 6ab 16b c) a ab 5b a + ab 1b 9s + 0st + 5t d) 0s 9st + 0t 5s 5st 6t 1s + 5st 5t Apply, Solve, Communicate 9. Communication On a soccer pitch, the goal area or goal bo is inside the penalty area or penalty bo and forms part of it. The dimensions of the goal bo and the penalty bo can be represented as shown. a) Write an epression that represents the area of the goal bo. b) Write an epression that represents the area of the penalty bo. c) Determine how many times as great the area of the penalty bo is as the area of the goal bo. d) Does the fact that represents 16.5 m affect your answer to part c)? Eplain. B 10. Measurement Write the area of the rectangle in simplest form goal goal bo penalty bo 11. Measurement The area of the trapezoid is 6y 5y 6. What is its height? y y 1.6 Multiplying and Dividing Rational Epressions MHR 51

9 1. Measurement a) Write, but do not simplify, an epression for the area of ABC. b) Write, but do not simplify, an epression for the area of DEF. c) Write and simplify an epression that represents the ratio of the area of ABC to the area of DEF. B A C E D + 6 F 1. Application Write and simplify an epression that represents the fraction of the area of the large rectangle covered by the shaded rectangle a c 1. In divisions of the form, the epressions b, c, and d must all b d be eamined for possible restrictions on the variables. Eplain why. 15. Measurement Two rectangles have common sides with a right triangle, as shown. The areas and widths of the rectangles are as indicated. Write and simplify an epression for the area of the triangle. 1 A = C 16. As the value of y increases, what happens to the value of each of the following epressions? Eplain. 15y y 1 y 1 8y + y 1 8y a) b) 6y + 7y 10y y 1 6y y 9y Inquiry/Problem Solving Write two different pairs of rational + 7y + y epressions with a product of. y 18. Write four different pairs of rational epressions with a product of 8 +. Compare your epressions with a classmate s A = MHR Chapter 1