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Algebraic Expressions and Equations: 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. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form. This module contains the proficiency exam for the chapter "Algebraic Expressions and Equations".

Proficiency Exam

Exercise 1

((Reference)) In the expression below, specify the number of terms that are present, then list them.
 3a(a+1)−(a+2)(a−3) 3a(a+1)−(a+2)(a−3)

Solution

two:  3a( a+1 ),−( a+2 )( a−3 ) two:  3a( a+1 ),−( a+2 )( a−3 )

Exercise 2

((Reference)) List, if there are any, the common factors of
 20 x 3 y 2 +15 x 3 y 2 z 2 +10 x 3 z 2 20 x 3 y 2 +15 x 3 y 2 z 2 +10 x 3 z 2

Solution

5 x 3 5 x 3

Exercise 3

((Reference)) How many y 2 (b+2)'s y 2 (b+2)'s in 8x y 2 (b+2)(b−6) 8x y 2 (b+2)(b−6) ?

Solution

8x( b−6 ) 8x( b−6 )

Exercise 4

((Reference)) Write the coefficient of x 3 x 3 in 8 x 3 y 3 z 8 x 3 y 3 z .

Solution

8 y 3 z 8 y 3 z

Exercise 5

((Reference)) Find the value of P 2 P 2 if k=4 k=4 and a=3 a=3 .
 P 2 =k a 3 P 2 =k a 3

Solution

108

Exercise 6

((Reference)) Classify the polynomial given below as a monomial, binomial, trinomial, or none of these. Specify the degree of the polynomial and write the numerical coefficient of each term.
 3 x 3 y+4x y 4 +8 x 2 y 2 z 0 w,      z≠0 3 x 3 y+4x y 4 +8 x 2 y 2 z 0 w,      z≠0

Solution

trinomial; 5th degree;
numberical coefficients: 3, 4, 8

Simplify the algebraic expressions for the following problems.

Exercise 7

((Reference)) 4 x 2 +3x+2x+11 x 2 −3 4 x 2 +3x+2x+11 x 2 −3

Solution

15 x 2 +5x−3 15 x 2 +5x−3

Exercise 8

((Reference)) 3a[2(a+1)+4]−18a 3a[2(a+1)+4]−18a

Solution

6 a 2 6 a 2

Exercise 9

((Reference)) (x+2)(x+4) (x+2)(x+4)

Solution

x 2 +6x+8 x 2 +6x+8

Exercise 10

((Reference)) (3a−7)(2a+10) (3a−7)(2a+10)

Solution

6 a 2 +16a−70 6 a 2 +16a−70

Exercise 11

((Reference)) (y+3) 2 (y+3) 2

Solution

y 2 +6y+9 y 2 +6y+9

Exercise 12

((Reference)) (6a+7y) 2 (6a+7y) 2

Solution

36 a 2 +84ay+49 y 2 36 a 2 +84ay+49 y 2

Exercise 13

((Reference)) (4x−9y) 2 (4x−9y) 2

Solution

16 x 2 −72xy+81 y 2 16 x 2 −72xy+81 y 2

Exercise 14

((Reference)-(Reference)) 3 x 2 (2x+5)(3x+1) 3 x 2 (2x+5)(3x+1)

Solution

18 x 4 +51 x 3 +15 x 2 18 x 4 +51 x 3 +15 x 2

Exercise 15

((Reference)-(Reference)) (3a−b)(4a−3b) (3a−b)(4a−3b)

Solution

12 a 2 −13ab+3 b 2 12 a 2 −13ab+3 b 2

Exercise 16

((Reference)-(Reference)) −6 y 2 (2y+3 y 2 −4) −6 y 2 (2y+3 y 2 −4)

Solution

−18 y 4 −12 y 3 +24 y 2 −18 y 4 −12 y 3 +24 y 2

Exercise 17

((Reference)-(Reference)) −4 b 3 ( b 2 −1) 2 −4 b 3 ( b 2 −1) 2

Solution

−4 b 7 +8 b 5 −4 b 3 −4 b 7 +8 b 5 −4 b 3

Exercise 18

((Reference)-(Reference)) (2 a 3 +3 b 2 ) 2 (2 a 3 +3 b 2 ) 2

Solution

4 a 6 +12 a 3 b 2 +9 b 4 4 a 6 +12 a 3 b 2 +9 b 4

Exercise 19

((Reference)-(Reference)) 6a(a−2)−(2 a 2 +a−11) 6a(a−2)−(2 a 2 +a−11)

Solution

4 a 2 −13a+11 4 a 2 −13a+11

Exercise 20

((Reference)-(Reference)) (5h+2k)(5h−2k) (5h+2k)(5h−2k)

Solution

25 h 2 −4 k 2 25 h 2 −4 k 2

Exercise 21

((Reference)-(Reference)) Subtract 4 a 2 −10 4 a 2 −10 from 2 a 2 +6a+1 2 a 2 +6a+1 .

Solution

−2 a 2 +6a+11 −2 a 2 +6a+11

Exercise 22

((Reference)-(Reference)) Add three times 6x−1 6x−1 to two times −4x+5 −4x+5 .

Solution

10x+7 10x+7

Exercise 23

((Reference)-(Reference)) Evaluate 6 k 2 +2k−7 6 k 2 +2k−7 if k=−1 k=−1 .

Solution

−3 −3

Exercise 24

((Reference)-(Reference)) Evaluate −2m (m−3) 2 −2m (m−3) 2 if m=−4 m=−4 .

Solution

392

Exercise 25

((Reference)) What is the domain of y= 3x−7 x+3 y= 3x−7 x+3 ?

Solution

all real numbers except ( −3 ) all real numbers except ( −3 )

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