1.8 Adding and Subtracting Rational Expressions, II

Transcription

1 .8 Adding and Subtracting Rational Epressions, II The three-toed sloth of South America moves very slowly. It can travel twice as fast in a tree as it can on the ground. If its speed on the ground is s metres per minute, its speed in a tree is s metres per minute. The sloth can travel m on the ground in minutes and m in a tree in minutes. The total time it takes to s s travel m on the ground and m in a tree is + minutes. The epression s s + is the sum of the two rational epressions with different denominators. s s Adding these rational epressions involves finding the LCM of two monomials that include variables. INVESTIGATE & INQUIRE. To find the LCM of each group of monomials, copy and complete the table. a) b) c) d) Monomials a b 6b 0 y y 6yz 9z 8 y 0y Factored Form a a b b b LCM. Describe a method for mentally finding the LCM of monomials that include variables.. a) Factor the binomials + and + 6. b) Write the LCM of the binomials in factored form. 6 MHR Chapter

2 . Write the LCM of each pair of epressions in factored form. a) a 9, a b) +, + c) y 6, y 9 d) + + 6, +. The total time a three-toed sloth takes to travel m on the ground and m in a tree is + minutes. s s a) State the common denominator of the two rational epressions. b) Add the epressions. c) If s represents. m/min, what is the total time, in minutes, the sloth takes to travel m on the ground and m in a tree? EXAMPLE Adding and Subtracting With Monomial Denominators Simplify +. State the restriction on the variable. a a a SOLUTION Find the LCD. a = a a = a a a = a a a The LCD is a a a or 0a. a + a a a () a() 0() Write with the common denominator: = + a (a) a(a ) 0(a ) 8a a 0 = + 0a 0a 0a Add or subtract the numerators: = 8a a + 0 0a Eclude the values for whicha = 0 or a = 0, or a = 0. a = 0 a = 0 a = 0 So, a 0. 8a a + 0 Therefore, + =, a 0. a a a 0a.8 Adding and Subtracting Rational Epressions, II MHR 6

3 EXAMPLE Denominators With a Common Binomial Factor m Simplify +. State the restriction on the variable. m m 6 SOLUTION m = (m ) m 6 = (m ) The LCD is (m ) or 6(m ). m + m m 6 m = + (m ) (m ) (m) () Write with the common denominator: = + (m ) (m ) m 6 = + 6(m ) 6(m ) m 6 + 6(m ) Add or subtract the numerators: = 6(m ) m 6 + 6m Epand the numerator: = 6(m ) Simplify: = 9m 8 6(m ) 6(m ) 6(m ) 6(m )() 6(m )() Factor: = (m ) Divide by the common factors: = (m ) = (m ) (m ) The numerator and denominator have common factors, so simplify further. Eclude the values for which m = 0 or m 6 = 0. m = m = So, m. m Therefore, + =, m. m m 6 6 MHR Chapter

4 EXAMPLE Denominators With Different Binomial Factors Simplify +. State the restrictions on the variable SOLUTION = 6( + ) = ( ) The LCD is ( + )( ) = + 6( + ) ( ) ( ) ( + ) Write with the common denominator: = + ( ) 6( + ) ( + ) ( ) ( ) ( + ) = + ( + )( ) ( + )( ) ( ) + ( + ) Add the numerators: = ( + )( ) Epand the numerator: = ( + )( ) Simplify: = ( + )( ) Eclude the values for which = 0 or = 0. = = So,, Therefore, + =,, ( + )( ) EXAMPLE Trinomial Denominators Simplify. State the restrictions on the variable. y + y + 6 y y SOLUTION y + y + 6 = (y + )(y + ) y y = (y + )(y ) The LCD is (y + )(y + )(y )..8 Adding and Subtracting Rational Epressions, II MHR 6

5 y + y + 6 y y = (y + )(y + ) (y + )(y ) y y + Write with the common denominator: = y ( y + )( y + ) y + ( y ) = (y + )( y + )( y ) ( y ) ( y + ) Subtract the numerators: = ( y + )( y + )( y ) Epand the numerator: = Simplify: = Eclude the values for which (y + )(y + ) = 0 or (y + )(y ) = 0. y + = 0 or y + = 0 y + = 0 or y = 0 y = y = y = y = So, y,,. y 6 Therefore, =, y,,. y + y + 6 y y ( y + )( y + )( y ) Key Concepts 66 MHR Chapter y 6 y 0 (y + )( y + )( y ) y 6 ( y + )( y + )( y ) ( y + )( y ) ( y + ) ( y + )( y + )( y ) To add or subtract rational epressions with a common polynomial denominator, write the numerators over the common denominator, and add or subtract the numerators. To add or subtract rational epressions with different polynomial denominators, rewrite the epressions with a common denominator. Then, write the numerators over the common denominator, and add or subtract the numerators. Communicate Your Understanding 7. a) Describe how you would simplify +. b) What is the restriction on the variable?. a) Describe how you would simplify +. b) What are the restrictions on the variable?

6 Practise In each of the following, state any restrictions on the variables. A. Write an equivalent epression with a denominator of y. a) b) y y y c) d) y 6. Find the LCM. a) 0a b, ab b) m n, mn, 6mn c), 6y, y d) 0s t, 0s t, st. Simplify. a) + b) + y y y c) + d) + m n m n e) + f) m + mn m g) + + mn + y + y y h) y y. Find the LCM of each of the following. Leave answers in factored form. a) m + 6, m + b) y, y + 0 c) m 8, 6m 8 d) 8, 0. Simplify. a) + b) + + y c) t t d) + t t m + e) f) + y 8 y 6 a + 6. Simplify. a) m + b) + + m m + c) + d) + + t e) f) + n n + g) t + h) 8 t + 6t i) s m m j) + s s m m 8 7. State the LCM in factored form. a) +, + + b) y + 6y + 8, y c) t t, t t d), e) m + 6m + 9, m m 8. Simplify. a) y b) y 6 y + c) + y m + 6a +.8 Adding and Subtracting Rational Epressions, II MHR 67

7 a d) a 7a + e) f) n a a + 6n n + 9. Simplify. a) + m + m + m + 7m + b) + + a c) a a 9a + 0 m m d) + m 9m + 8 m m + 0 e) y y f) y 9 y y Simplify. t + a) + t t t + y + y b) + y y + y + c) n + n n d) + n 6 n m + m e) m m m m + a + a f) a a + a + w w g) + w + w + w + w 8 + h) z z + i) z z 8z Apply, Solve, Communicate B. Inquiry/Problem Solving a) Copy and complete the table. The first line has been completed. Epressions,, 8 y, 9y a +, a t, t Product LCM b) How is the product LCM GCF related to the product of each pair of epressions? c) Eplain why the relationship you found in part b) eists. GCF LCM GCF 68 MHR Chapter

8 . Application a) An RCMP patrol boat left Goderich and travelled for km along the coast of Lake Huron at a speed of s kilometres per hour. Write an epression that represents the time taken, in hours. b) The boat returned to Goderich at a speed of s kilometres per hour. Write an epression that represents the time taken, in hours. c) Write and simplify an epression that represents the total time, in hours, the boat was travelling. d) If s represents 0 km/h, for how many hours was the boat travelling?. Communication Write a problem that satisfies the following conditions. Have a classmate solve your problem. simplifies using addition and/or subtraction includes three rational epressions with different denominators that contain variables has the LCD as the denominator of one of the rational epressions. Simplify. m + m + m a) + b) m + m + m y y + y c) + d) 7 + y y 6y + 8 z z + z e) + f) z z C. Write two rational epressions with binomial denominators and with each of the following sums. Compare your answers with a classmate s. + 8 a) b) ( + )( + ) c) d) ( )( ) 9 A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application Suppose you drive an average of km/year. With your present car, you can drive 0 km per litre of fuel. You are thinking of buying a new car that you could drive km farther per litre of fuel. Fuel currently costs $0.68/L. If the new car would save you $.80 in yearly fuel costs, find the number of kilometres you could drive the new car per litre of fuel..8 Adding and Subtracting Rational Epressions, II MHR 69