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Rational Expressions: Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

This module contains the proficiency exam for the chapter "Rational Expressions".

Proficiency Exam

Exercise 1

((Reference)) Find the domain of 5a+1 a 2 −5a−24 . 5a+1 a 2 −5a−24 .

Solution

a≠−3,8 a≠−3,8

Exercise 2

For the following problems, fill in the missing term.
((Reference)) − 3 x+4 = x+4 − 3 x+4 = x+4

Solution

−3 −3

Exercise 3

((Reference)) 2x+5 −x+1 = x−1 2x+5 −x+1 = x−1

Solution

−2x−5 −2x−5

Exercise 4

For the following problems, reduce to lowest terms.
((Reference)) 30 x 6 y 3 ( x−3 ) 2 ( x+5 ) 2 6x y 3 ( x+5 ) 30 x 6 y 3 ( x−3 ) 2 ( x+5 ) 2 6x y 3 ( x+5 )

Solution

5 x 5 ( x−3 ) 2 ( x+5 ) 5 x 5 ( x−3 ) 2 ( x+5 )

Exercise 5

((Reference)) x 2 +10x+24 x 2 +x−30 x 2 +10x+24 x 2 +x−30

Solution

x+4 x−5 x+4 x−5

Exercise 6

((Reference)) 8 x 2 +2x−3 4 x 2 +12x−7 8 x 2 +2x−3 4 x 2 +12x−7

Solution

4x+3 2x+7 4x+3 2x+7

Exercise 7

((Reference)) Replace N N with the proper quantity.
 x+2 x−1 = N x 2 −4x+3 x+2 x−1 = N x 2 −4x+3

Solution

( x−3 )( x+2 ) ( x−3 )( x+2 )

Exercise 8

((Reference)) Assume that a 2 +a−6, a 2 +a−6, a 2 −a−12, a 2 −a−12, and a 2 −2a−8 a 2 −2a−8 are denominators of rational expressions. Find the LCD.

Solution

( a+2 )( a−2 )( a+3 )( a−4 ) ( a+2 )( a−2 )( a+3 )( a−4 )

Exercise 9

For the following problems, perform the operations.
((Reference)) 3a+4 a+6 − 2a−1 a+6 3a+4 a+6 − 2a−1 a+6

Solution

a+5 a+6 a+5 a+6

Exercise 10

((Reference)) 18 x 3 y 5 a 2  ·  15 a 3 b 6 x 2 y 18 x 3 y 5 a 2  ·  15 a 3 b 6 x 2 y

Solution

9abx 9abx

Exercise 11

((Reference)) y 2 −y−12 y 2 +3y+2  ·  y 2 +10y+16 y 2 −7y+12 y 2 −y−12 y 2 +3y+2  ·  y 2 +10y+16 y 2 −7y+12

Solution

( y+3 )( y+8 ) ( y+1 )( y−3 ) ( y+3 )( y+8 ) ( y+1 )( y−3 )

Exercise 12

((Reference)) y−2 y 2 −11y+24 + y+4 y 2 +3y−18 y−2 y 2 −11y+24 + y+4 y 2 +3y−18

Solution

2( y 2 −22 ) ( y−8 )( y−3 )( y+6 ) 2( y 2 −22 ) ( y−8 )( y−3 )( y+6 )

Exercise 13

((Reference)) 9 2x+7 + 4 6x−1 9 2x+7 + 4 6x−1

Solution

62x+19 ( 2x+7 )( 6x−1 ) 62x+19 ( 2x+7 )( 6x−1 )

Exercise 14

((Reference)) 16 x 5 ( x 2 −1 ) 9x−9 ÷ 2 x 2 −2x 3 16 x 5 ( x 2 −1 ) 9x−9 ÷ 2 x 2 −2x 3

Solution

8 x 4 ( x+1 ) 3( x−1 ) 8 x 4 ( x+1 ) 3( x−1 )

Exercise 15

((Reference)) ( m+3 )÷ 2m+6 5m+1 ( m+3 )÷ 2m+6 5m+1

Solution

5m+1 2 5m+1 2

Exercise 16

((Reference)) 3y+10 8 y 2 +10y−3 − 5y−1 4 y 2 +23y−6 3y+10 8 y 2 +10y−3 − 5y−1 4 y 2 +23y−6

Solution

−7 y 2 +15y+63 ( 4y−1 )( 2y+3 )( y+6 ) −7 y 2 +15y+63 ( 4y−1 )( 2y+3 )( y+6 )

Exercise 17

((Reference)) Solve 1 x+3 + 3 x−3 = x x 2 −9 . 1 x+3 + 3 x−3 = x x 2 −9 .

Solution

x=−2 x=−2

Exercise 18

((Reference)) Solve 12 m−4 +5= 3m m−4 . 12 m−4 +5= 3m m−4 .

Solution

No solution; m=4 m=4 is excluded.

Exercise 19

((Reference)) When the same number is added to both the numerator and denominator of the fraction 5 3 , 5 3 , the result is 6 5 . 6 5 . What is the number that is added?

Solution

7

Exercise 20

((Reference)) Person A, working alone, can complete a job in 20 hours. Person B, working alone, can complete the same job in 30 hours. How long will it take both people, working together, to complete the job?

Solution

12 hours

Exercise 21

((Reference)) The width of a rectangle is 1 foot longer than one half the length. Find the dimensions (lengh and width) of the rectangle if the perimeter is 44 feet.

Solution

8 ft by 14 ft

Exercise 22

((Reference)) Simplify the complex fraction 4− 3 x 4+ 3 x . 4− 3 x 4+ 3 x .

Solution

4x−3 4x+3 4x−3 4x+3

Exercise 23

((Reference)) Simplify the complex fraction 1− 5 x − 6 x 2 1+ 6 x + 5 x 2 . 1− 5 x − 6 x 2 1+ 6 x + 5 x 2 .

Solution

x−6 x+5 x−6 x+5

Exercise 24

((Reference)) Perform the division: x 3 +10 x 2 +21x−18 x+6 . x 3 +10 x 2 +21x−18 x+6 .

Solution

x 2 +4x−3 x 2 +4x−3

Exercise 25

((Reference)) Perform the division: 2 x 3 +5x−1 x−2 . 2 x 3 +5x−1 x−2 .

Solution

2 x 2 +4x+13+ 25 x−2 2 x 2 +4x+13+ 25 x−2

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