4.4 Rational Expressions

Transcription

1 4.4 Rational Epressions Learning Objectives Simplify rational epressions. Find ecluded values of rational epressions. Simplify rational models of real-world situations. Introduction A rational epression is reduced to lowest terms by factoring the numerator and denominator completely and divide out common factors. For eample, the epression z 1 y z 1 = y simplifies to simplest form by dividing out the common factor z. Simplify Rational Epressions. To simplify rational epressions means that the numerator and denominator of the rational epression have no common factors. In order to simplify to lowest terms, we factor the numerator and denominator as much as we can and divide out common factors from the numerator and the denominator of the fraction. Eample 1 Simplify each rational epression. a) b) c) a) Factor the numerator and denominator completely. Divide the common term (2 1). Answer (2 1) (2 1)(+1) b) Factor the numerator and denominator completely. ( 1)( 1) 8( 1) Divide the common term ( 1). Answer 1 8 c) Factor the numerator and denominator completely. ( 2)(+2) ( 2)( 3) Divide the common term ( 2). Answer +2 3 Common mistakes in simplifying rational epressions: When simplify rational epressions, you are only allowed to divide out common factors from the denominator but NOT common terms. For eample, in the epression 261

2 4.4. Rational Epressions we can cross out the ( 3) factor because ( 3) ( 3) = 1. We write ( + 1).( 3) ( + 2).( 3) ( + 1) ( 3) 1 ( + 1) = ( + 2) ( 3) 1 ( + 2) However, don t make the mistake of dividing out common terms in the numerator and denominator. For instance, in the epression. we cannot cross out the 2 terms When we cross out terms that are part of a sum or a difference we are violating the order of operations (PEMDAS). We must remember that the fraction sign means division. When we perform the operation we are dividing the numerator by the denominator ( 2 + 1) ( 2 5) ( 2 + 1) ( 2 5) The order of operations says that we must perform the operations inside the parenthesis before we can perform the division. Try this with numbers: = 10 4 = 2.5 But if we divide incorrectly we obtain the following = 1 5 = 0.2. CORRECT INCORRECT Find Ecluded Values of Rational Epressions Whenever a variable epression is present in the denominator of a fraction, we must be aware of the possibility that the denominator could be zero. Since division by zero is undefined, certain values of the variable must be ecluded. 262

3 These values are the vertical asymptotes (i.e. values that cannot eist for ). For eample, in the epression ( 2 3), the value of = 3 must be ecluded. To find the ecluded values we simply set the denominator equal to zero and solve the resulting equation. Eample 2 Find the ecluded values of the following epressions. a) +4 b) c) 2 5 a) When we set the denominator equal to zero we obtain. + 4 = 0 = 4 is the ecluded value. b) When we set the denominator equal to zero we obtain. 2 6 = 0 Solve by factoring. ( 3)( + 2)= 0 = 3 and = 2 are the ecluded values. c) When we set the denominator equal to zero we obtain. 2 5 = 0 Solve by factoring. ( 5)=0 = 0 and = 5 are the ecluded values. Removable Zeros Notice that in the epressions in Eample 1, we removed a division by zero when we simplified the problem. For instance, was rewritten as (2 1) (2 1)( + 1). This epression eperiences division by zero when = 1 2 and = 1. However, when we divide out common factors, we simplify the epression to The reduced form allows the value = 1 2. We thus removed a division by zero and the reduced epression has only = 1 as the ecluded value. Technically the original epression and the simplified epression are not the same. When we simplify to simplest form we should specify the removed ecluded value. Thus, The epression from Eample 1, part b simplifies to = 2 + 1, = 1 8, 1 263

4 4.4. Rational Epressions The epression from Eample 1, part c simplifies to = + 2 3, 2 Simplify Rational Models of Real-World Situations Many real world situations involve epressions that contain rational coefficients or epressions where the variable appears in the denominator. Eample 3 The gravitational force between two objects in given by the formula F = G m 1m 2. if the gravitation constant is given by G (N m 2 /kg 2 ). The force of attraction between the Earth and the Moon is F = N (with masses of m 1 = kg for the Earth and m 2 = kg for the Moon). What is the distance between the Earth and the Moon? Lets start with the Law of Gravitation formula. F = G m 1m 2 Now plug in the known values N m2 N = kg 2.( kg)( kg) Multiply the masses together N m2 N = kg kg 2 Divide out the kg 2 units N m2 N = kg kg 2 Multiply the numbers in the numerator N = N m 2 Multiply both sides by N = N m 2 Divide out common factors N = N m 2 d2 Simplify N = N m 2 Divide both sides by N N m 2 = N Simplify. = m 2 Take the square root of both sides. d = m Answer The distance is m. This is indeed the distance between the Earth and the Moon. Eample 4 The area of a circle is given by A = πr 2 and the circumference of a circle is given by C = 2πr. Find the ratio of the circumference and area of the circle. The ratio of the circumference and area of the circle is: 2πr πr 2 264

5 We divide out common factors from the numerator and denominator. 2 π 1 r 1 π 1 r 2 r Simplify. Answer 2 r Eample 5 The height of a cylinder is 2 units more than its radius. Find the ratio of the surface area of the cylinder to its volume. Define variables. Let R = the radius of the base of the cylinder. Then, R + 2 = the height of the cylinder To find the surface area of a cylinder, we need to add the areas of the top and bottom circle and the area of the curved surface. The volume of a cylinder is the area of the base of the cylinder times its height, so: The volume of the cylinder is V = The ratio of the surface area of the cylinder to its volume is Distribute to eliminate the parentheses in the numerator. Combine like terms in the numerator. Factor common terms in the numerator. Divide out common terms in the numerator and denominator. Simplify. 2πR 2 + 2πR(R + 2) 2πR 2 + 2πR 2 + 4πR 4πR 2 + 4πR 4πR(R + 1) 4 π 1 R 1 (R + 1) π 1 R 2 R(R + 2) 4(R + 1) R(R + 2) Answer 265

6 4.4. Rational Epressions Review Questions Simplify each rational epression to lowest terms Find the ecluded values for each rational epression ( 1) Suppose that two objects attract each other with a gravitational force of 20 Newtons. If the distance between the two objects is doubled, what is the new force of attraction between the two objects? 26. Suppose that two objects attract each other with a gravitational force of 36 Newtons. If the mass of both objects was doubled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects? 27. A sphere with radius r has a volume of 4 3 πr3 and a surface area of 4πr 2. Find the ratio the surface area to the volume of a sphere. 28. The side of a cube is increased by a factor of two. Find the ratio of the old volume to the new volume. 29. The radius of a sphere is decreased by four units. Find the ratio of the old volume to the new volume. Review Answers , , , 0

7 , 3, 4, , , , 5, = = = = 2, = none 18. = 4, = = 0, = = 0, = 5, = , , none , Newtons Newtons r r 3 (r 4) 3 267