1.5 Simplifying Rational Expressions

Transcription

1 .5 Simplifying Rational Epressions Canada officially has two national games, lacrosse and hockey. Lacrosse is thought to have originated with the Algonquin tribes in the St. Lawrence Valley. The game was very popular in the late nineteenth century and was at one time an Olympic sport. Canadian lacrosse teams won gold medals at the Summer Olympics in 904 and 908. There are two forms of lacrosse bo lacrosse, which is played indoors, and field lacrosse. When field lacrosse is played under international rules, the width of the rectangular field can be represented by and the area of the field by the polynomial 0. Thus, the length of the field can be represented by 0 0 the epression. This is an eample of a rational epression, which is a quotient whose numerator and denominator are polynomials. The following are also rational epressions. + y y y a + b a b INVESTIGATE & INQUIRE. a) Factor from the epression for the area of a lacrosse field, 0. b) Record the other factor and eplain why it represents the length of the field. c) Describe how you could simplify the other epression for the length, 0, to give the same epression as in part b).. The rectangle shown has a width of y and an area of y + 6y. a) Factor y from the epression for the area. b) Record the other factor and eplain why it represents the length. y y + 6y.5 Simplifying Rational Epressions MHR 5

2 c) Use the width and the area to write a rational epression that represents the length. d) Describe how you could simplify the rational epression from part c) to give the same epression as in part b).. Use your results from questions and to write a rule for simplifying a rational epression in which the denominator is a monomial factor of the numerator. 4. Use your rule to simplify each of the following. 4t + 8t 0m + 5m + 5m 6r 4 r + 6r a) b) c) 4t 5m r 5. The epressions from question represent the dimensions of a lacrosse field for both the women s and the men s games. a) For the women s game, played -a-side, represents 60 m. What are the dimensions of the field, in metres? b) For the men s game, played 0-a-side, represents 55 m. What are the dimensions of the field, in metres? EXAMPLE Monomial Denominator Simplify. State the restriction on the variable. 6 Factor the numerator: = (4 + + ) 6 Divide by the common factor, 6: = 6 (4 + + ) 6 = Division by 0 is not defined, so eclude values of for which 6 = 0. 6 = 0 when = 0, so Therefore, = 4 + +, 0 6 Ecluded values are known as restrictions on the variable. 6 MHR Chapter

3 The solution to Eample could have been found by another method, since the distributive property also applies to division. + For eample, = 7 7 = So, = = 4 + +, 0 EXAMPLE Binomial Denominator Epress 4 in simplest form. State the restrictions on the variable. 4 Factor the denominator: = Divide by the common factor, : = = ( ) Eclude values of for which 4 = 0. 4 = ( ), so 4 = 0 when ( ) = 0 = 0 or = 0 = 0 or = Therefore, =, 0,. 4 ( ) ( ) ( ).5 Simplifying Rational Epressions MHR 7

4 EXAMPLE Removing a Common Factor of Simplify. State any restrictions on the variable Factor the denominator: = Factor from the numerator: = ( ) ( ) ( ) Divide by the common factor, ( ): = ( ) ( ) = or Eclude values of for which 4 6 = = 0 when ( ) = 0. = 0 = Therefore, =,. 4 6 EXAMPLE 4 Trinomial Numerator and Denominator Epress MHR Chapter in simplest form. State the restrictions on the variable. Factor the numerator and the denominator: = ( + 5)( ) ( + 5)( + ) ( + 5) ( ) Divide by the common factor, ( + 5): = ( + 5) ( + ) = +

5 Eclude values of for which = = ( + 5)( + ), so = 0 when ( + 5)( + ) = = 0 or + = 0 = 5 or = + 0 Therefore, =, 5, EXAMPLE 5 Trinomial Numerator and Denominator y y 5 Simplify. State the restrictions on the variable. 4y y + y y 5 4y y + (y )(y + 5) Factor the numerator and denominator: = (4y )(y ) (y ) (y + 5) Divide by the common factor, ( y ): = (4y )(y ) y + 5 = 4y Eclude values of y for which 4y y + = 0. 4y y + = (4y )(y ), so 4y y + = 0 when (4y )( y ) = 0 4y = 0 or y = 0 y = or y = 4 y y 5 y + 5 Therefore, =, y,. 4y y + 4y 4 Key Concepts To simplify rational epressions, a) factor the numerator and the denominator b) divide by common factors To state the restriction(s) on the variable in a rational epression, determine and eclude the value(s) of the variable that make the denominator 0..5 Simplifying Rational Epressions MHR 9

6 Communicate Your Understanding + 4. Eplain why is a restriction on the variable for the epression.. Describe how you would simplify a) Describe how you would simplify. b) Describe how you would determine the restrictions on the variable. 4. Write an epression in one variable for the denominator of a rational epression, if the restrictions on the variable are,. Practise In each of the following, state any restrictions on the variables. A. Simplify. 0 4 g) h) t + 6t 5t 6a + 9a a) b) t a y i) 0y 4 + 5y 5y 6 y y c) 5y. Simplify. 4n 4 4n + 6n + 8n d) 6t 6 4m + 4 n a) b) t 6 8m 4 4m 8mn 6 y e) f) 5 0 a + a 4mn 8 y c) d) 6 a a 6a bc 4 4 y z g) h) a b c 0 y z e) f) m(m 4) i) 4 + 4y 4a b + 8ab 7m g) h) 5 + 5y 6a 6a. Epress in simplest form. 5y + 0 i) 5 8t (t + 5) y + 4y a) b) 5( + 4) 4t(t 5) 4. Epress in simplest equivalent form. 7( ) (m )(m + ) c) d) m y + 0y ( ) (m + 4)(m ) a) b) m 5m + 6 y + 5 y e) f) + 6 r y + y c) d) 6 7 5r MHR Chapter

7 a + a 9 e) f) a + a + y 6y w + t 8t + 4 g) h) w + w + 6t 4t 8z + 6z 5 + y y i) j) 9z 6 + y 5. Simplify. y a) b) y t 6 0w c) d) 4 8t 5w 9 4y e) f) 8y 6. Simplify a a a) b) a 9a + 0 m 5m + 6 y 8y + 5 c) d) m + m 5 y n n e) f) + 6 n + n 6 p + 8p + 6 t t g) h) p 6 t t + 6v + v i) j) 4v + 8v z 7z + m mn n k) l) 9z 6z + 4m 4mn n Apply, Solve, Communicate 7. Saskatchewan flag The area of a Saskatchewan flag can be represented by the polynomial + + and its width by +. a) Write a rational epression that represents the length. b) Write the epression in simplest form. c) If represents unit of length, what is the ratio length:width for a Saskatchewan flag? B 8. Simplify, if possible. y + a) b) c) + y t 7 t s + d) e) f) t 7 (s + t) For which values of are the following rational epressions not defined? y 4 a) b) c) y + y + d) e) f) + 5y + y 8 4 9y.5 Simplifying Rational Epressions MHR 4

8 0. Communication State whether each of the following rational + epressions is equivalent to the epression. Eplain a) b) c) ( + ) + d) e) f) ( ). Cube For a cube of edge length, find the ratio of the volume to the surface area. Simplify, if possible.. Application For a sphere of radius r, find the ratio of the volume to the surface area. Simplify, if possible.. Pattern The first 4 diagrams of two patterns are shown. Pattern Pattern n = 4 n = 4 a) For pattern, epress the number of asterisks in the nth diagram in terms of n. b) For pattern, the number of asterisks in the nth diagram is given by the binomial product (n + )(n + ), where and represent whole numbers. Replace and in the binomial product by their correct values. c) Divide your polynomial from part b) by your epression from part a). d) Use your result from part c) to calculate how many times as many asterisks there are in the 0th diagram of pattern as there are in the 0th diagram of pattern. e) If a diagram in pattern has 0 asterisks, how many asterisks are in the corresponding diagram of pattern? f) If a diagram in pattern has 95 asterisks, how many asterisks are in the corresponding diagram in pattern? 4 MHR Chapter

9 4. Measurement Find the ratio of the area of the square to the area of the trapezoid. Simplify, if possible Rectangular prism Find the ratio of the volume to the surface area for the rectangular prism shown. Simplify, if possible C 6. Write rational epressions in one variable so that the restrictions on the variables are as follows. a) b) y 0, c) a, d) t, ± 4 7. Technology a) Use a graphing calculator to graph the equations + y = and y = + in the same standard viewing window. Eplain your observations. b) Display the tables of values for the two equations. Compare and eplain the values of y when = Inquiry/Problem Solving a) For a solid cone with radius r, height h, and slant height s, find the ratio of the volume to the surface area. Simplify, if possible. b) Determine whole-number values of r, h, and s that give the ratio in part a) a numerical value of. WORD Power Lewis Carroll invented a word game called doublets. The object of the game is to change one word to another by changing one letter at a time. You must form a real word each time you change a letter. The best solution has the fewest steps. Change the word RING to the word BELL by changing one letter at a time Simplifying Rational Epressions MHR 4