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Algebraic Expressions and Equations: Exercise Supplement

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 provides an exercise supplement for the chapter "Algebraic Expressions and Equations".

Exercise Supplement

Algebraic Expressions ((Reference))

For the following problems, write the number of terms that appear, then write the terms.

Exercise 1

4 x 2 +7x+12 4 x 2 +7x+12

Solution

three: 4 x 2 ,7x,12 three: 4 x 2 ,7x,12

Exercise 2

14 y 6 14 y 6

Exercise 3

c+8 c+8

Solution

two: c,8 two: c,8

Exercise 4

8 8

List, if any should appear, the common factors for the following problems.

Exercise 5

a 2 +4 a 2 +6 a 2 a 2 +4 a 2 +6 a 2

Solution

a 2 a 2

Exercise 6

9 y 4 −18 y 4 9 y 4 −18 y 4

Exercise 7

12 x 2 y 3 +36 y 3 12 x 2 y 3 +36 y 3

Solution

12 y 3 12 y 3

Exercise 8

6(a+4)+12(a+4) 6(a+4)+12(a+4)

Exercise 9

4(a+2b)+6(a+2b) 4(a+2b)+6(a+2b)

Solution

2( a+2b ) 2( a+2b )

Exercise 10

17 x 2 y(z+4)+51y(z+4) 17 x 2 y(z+4)+51y(z+4)

Exercise 11

6 a 2 b 3 c+5 x 2 y 6 a 2 b 3 c+5 x 2 y

Solution

no common factors

For the following problems, answer the question of how many.

Exercise 12

x's in 9x? x's in 9x?

Exercise 13

(a+b)'s in 12(a+b)? (a+b)'s in 12(a+b)?

Solution

12

Exercise 14

a 4 's in 6 a 4 ? a 4 's in 6 a 4 ?

Exercise 15

c 3 's in 2 a 2 b c 3 ? c 3 's in 2 a 2 b c 3 ?

Solution

2 a 2 b 2 a 2 b

Exercise 16

(2x+3y) 2 's in 5(x+2y) (2x+3y) 3 ? (2x+3y) 2 's in 5(x+2y) (2x+3y) 3 ?

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.

Exercise 17

8z, z 8z, z

Solution

8

Exercise 18

16 a 3 b 2 c 4 ,  c 4 16 a 3 b 2 c 4 ,  c 4

Exercise 19

7y(y+3), 7y 7y(y+3), 7y

Solution

( y+3 ) ( y+3 )

Exercise 20

(−5) a 5 b 5 c 5 , bc (−5) a 5 b 5 c 5 , bc

Equations ((Reference))

For the following problems, observe the equations and write the relationship being expressed.

Exercise 21

a=3b a=3b

Solution

The value of a is equal to three time the value of b. The value of a is equal to three time the value of b.

Exercise 22

r=4t+11 r=4t+11

Exercise 23

f= 1 2 m 2 +6g f= 1 2 m 2 +6g

Solution

The value of f is equal to six times g more then one half time the value of m squared. The value of f is equal to six times g more then one half time the value of m squared.

Exercise 24

x=5 y 3 +2y+6 x=5 y 3 +2y+6

Exercise 25

P 2 =k a 3 P 2 =k a 3

Solution

The value of P squared is equal to the value of a cubed times k. The value of P squared is equal to the value of a cubed times k.

Use numerical evaluation to evaluate the equations for the following problems.

Exercise 26

C=2πr. Find C if π is approximated by  3.14 and r=6. C=2πr. Find C if π is approximated by  3.14 and r=6.

Exercise 27

I= E R . Find I if E=20 and R=2. I= E R . Find I if E=20 and R=2.

Solution

10

Exercise 28

I=prt. Find I if p=1000, r=0.06, and t=3. I=prt. Find I if p=1000, r=0.06, and t=3.

Exercise 29

E=m c 2 . Find E if m=120 and c=186,000. E=m c 2 . Find E if m=120 and c=186,000.

Solution

4.1515× 10 12 4.1515× 10 12

Exercise 30

z= x−u s . Find z if x=42, u=30, and s=12. z= x−u s . Find z if x=42, u=30, and s=12.

Exercise 31

R= 24C P(n+1) . Find R if C=35, P=300, and n=19. R= 24C P(n+1) . Find R if C=35, P=300, and n=19.

Solution

7 50  or 0.14 7 50  or 0.14

Classification of Expressions and Equations ((Reference))

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

Exercise 32

2a+9 2a+9

Exercise 33

4 y 3 +3y+1 4 y 3 +3y+1

Solution

trinomial; cubic; 4, 3, 1

Exercise 34

10 a 4 10 a 4

Exercise 35

147 147

Solution

monomial; zero; 147

Exercise 36

4xy+2y z 2 +6x 4xy+2y z 2 +6x

Exercise 37

9a b 2 c 2 +10 a 3 b 2 c 5 9a b 2 c 2 +10 a 3 b 2 c 5

Solution

binomial; tenth; 9, 10

Exercise 38

(2x y 3 ) 0 , x y 3 ≠0 (2x y 3 ) 0 , x y 3 ≠0

Exercise 39

Why is the expression 4x 3x−7 4x 3x−7 not a polynomial?

Solution

. . . because there is a variable in the denominator

Exercise 40

Why is the expression 5 a 3/4 5 a 3/4 not a polynomial?

For the following problems, classify each of the equations by degree. If the term linear, quadratic, or cubic applies, use it.

Exercise 41

3y+2x=1 3y+2x=1

Solution

linear

Exercise 42

4 a 2 −5a+8=0 4 a 2 −5a+8=0

Exercise 43

y−x−z+4w=21 y−x−z+4w=21

Solution

linear

Exercise 44

5 x 2 +2 x 2 −3x+1=19 5 x 2 +2 x 2 −3x+1=19

Exercise 45

(6 x 3 ) 0 +5 x 2 =7 (6 x 3 ) 0 +5 x 2 =7

Solution

quadratic

Combining Polynomials Using Addition and Subtraction ((Reference)) - Special Binomial Products ((Reference))

Simplify the algebraic expressions for the following problems.

Exercise 46

4 a 2 b+8 a 2 b− a 2 b 4 a 2 b+8 a 2 b− a 2 b

Exercise 47

21 x 2 y 3 +3xy+ x 2 y 3 +6 21 x 2 y 3 +3xy+ x 2 y 3 +6

Solution

22 x 2 y 3 +3xy+6 22 x 2 y 3 +3xy+6

Exercise 48

7(x+1)+2x−6 7(x+1)+2x−6

Exercise 49

2(3 y 2 +4y+4)+5 y 2 +3(10y+2) 2(3 y 2 +4y+4)+5 y 2 +3(10y+2)

Solution

11 y 2 +38y+14 11 y 2 +38y+14

Exercise 50

5[3x+7(2 x 2 +3x+2)+5]−10 x 2 +4(3 x 2 +x) 5[3x+7(2 x 2 +3x+2)+5]−10 x 2 +4(3 x 2 +x)

Exercise 51

8{3[4 y 3 +y+2]+6( y 3 +2 y 2 )}−24 y 3 −10 y 2 −3 8{3[4 y 3 +y+2]+6( y 3 +2 y 2 )}−24 y 3 −10 y 2 −3

Solution

120 y 3 +86 y 2 +24y+45 120 y 3 +86 y 2 +24y+45

Exercise 52

4 a 2 b c 3 +5ab c 3 +9ab c 3 +7 a 2 b c 2 4 a 2 b c 3 +5ab c 3 +9ab c 3 +7 a 2 b c 2

Exercise 53

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

Solution

5 x 2 +2x+3 5 x 2 +2x+3

Exercise 54

4k(3 k 2 +2k+6)+k(5 k 2 +k)+16 4k(3 k 2 +2k+6)+k(5 k 2 +k)+16

Exercise 55

2{5[6(b+2a+ c 2 )]} 2{5[6(b+2a+ c 2 )]}

Solution

60 c 2 +120a+60b 60 c 2 +120a+60b

Exercise 56

9 x 2 y(3xy+4x)−7 x 3 y 2 −30 x 3 y+5y( x 3 y+2x) 9 x 2 y(3xy+4x)−7 x 3 y 2 −30 x 3 y+5y( x 3 y+2x)

Exercise 57

3m[5+2m(m+6 m 2 )]+m( m 2 +4m+1) 3m[5+2m(m+6 m 2 )]+m( m 2 +4m+1)

Solution

36 m 4 +7 m 3 +4 m 2 +16m 36 m 4 +7 m 3 +4 m 2 +16m

Exercise 58

2r[4(r+5)−2r−10]+6r(r+2) 2r[4(r+5)−2r−10]+6r(r+2)

Exercise 59

abc(3abc+c+b)+6a(2bc+b c 2 ) abc(3abc+c+b)+6a(2bc+b c 2 )

Solution

3 a 2 b 2 c 2 +7ab c 2 +a b 2 c+12abc 3 a 2 b 2 c 2 +7ab c 2 +a b 2 c+12abc

Exercise 60

s 10 (2 s 5 +3 s 4 +4 s 3 +5 s 2 +2s+2)− s 15 +2 s 14 +3s( s 12 +4 s 11 )− s 10 s 10 (2 s 5 +3 s 4 +4 s 3 +5 s 2 +2s+2)− s 15 +2 s 14 +3s( s 12 +4 s 11 )− s 10

Exercise 61

6 a 4 ( a 2 +5) 6 a 4 ( a 2 +5)

Solution

6 a 6 +30 a 4 6 a 6 +30 a 4

Exercise 62

2 x 2 y 4 (3 x 2 y+4xy+3y) 2 x 2 y 4 (3 x 2 y+4xy+3y)

Exercise 63

5 m 6 (2 m 7 +3 m 4 + m 2 +m+1) 5 m 6 (2 m 7 +3 m 4 + m 2 +m+1)

Solution

10 m 13 +15 m 10 +5 m 8 +5 m 7 +5 m 6 10 m 13 +15 m 10 +5 m 8 +5 m 7 +5 m 6

Exercise 64

a 3 b 3 c 4 (4a+2b+3c+ab+ac+b c 2 ) a 3 b 3 c 4 (4a+2b+3c+ab+ac+b c 2 )

Exercise 65

(x+2)(x+3) (x+2)(x+3)

Solution

x 2 +5x+6 x 2 +5x+6

Exercise 66

(y+4)(y+5) (y+4)(y+5)

Exercise 67

(a+1)(a+3) (a+1)(a+3)

Solution

a 2 +4a+3 a 2 +4a+3

Exercise 68

(3x+4)(2x+6) (3x+4)(2x+6)

Exercise 69

4xy−10xy 4xy−10xy

Solution

−6xy −6xy

Exercise 70

5a b 2 −3(2a b 2 +4) 5a b 2 −3(2a b 2 +4)

Exercise 71

7 x 4 −15 x 4 7 x 4 −15 x 4

Solution

−8 x 4 −8 x 4

Exercise 72

5 x 2 +2x−3−7 x 2 −3x−4−2 x 2 −11 5 x 2 +2x−3−7 x 2 −3x−4−2 x 2 −11

Exercise 73

4(x−8) 4(x−8)

Solution

4x−32 4x−32

Exercise 74

7x( x 2 −x+3) 7x( x 2 −x+3)

Exercise 75

−3a(5a−6) −3a(5a−6)

Solution

−15 a 2 +18a −15 a 2 +18a

Exercise 76

4 x 2 y 2 (2x−3y−5)−16 x 3 y 2 −3 x 2 y 3 4 x 2 y 2 (2x−3y−5)−16 x 3 y 2 −3 x 2 y 3

Exercise 77

−5y( y 2 −3y−6)−2y(3 y 2 +7)+(−2)(−5) −5y( y 2 −3y−6)−2y(3 y 2 +7)+(−2)(−5)

Solution

−11 y 3 +15 y 2 +16y+10 −11 y 3 +15 y 2 +16y+10

Exercise 78

−[−(−4)] −[−(−4)]

Exercise 79

−[−(−{−[−(5)]})] −[−(−{−[−(5)]})]

Solution

−5 −5

Exercise 80

x 2 +3x−4−4 x 2 −5x−9+2 x 2 −6 x 2 +3x−4−4 x 2 −5x−9+2 x 2 −6

Exercise 81

4 a 2 b−3 b 2 −5 b 2 −8 q 2 b−10 a 2 b− b 2 4 a 2 b−3 b 2 −5 b 2 −8 q 2 b−10 a 2 b− b 2

Solution

−6 a 2 b−8 q 2 b−9 b 2 −6 a 2 b−8 q 2 b−9 b 2

Exercise 82

2 x 2 −x−(3 x 2 −4x−5) 2 x 2 −x−(3 x 2 −4x−5)

Exercise 83

3(a−1)−4(a+6) 3(a−1)−4(a+6)

Solution

−a−27 −a−27

Exercise 84

−6(a+2)−7(a−4)+6(a−1) −6(a+2)−7(a−4)+6(a−1)

Exercise 85

Add −3x+4 −3x+4 to 5x−8 5x−8 .

Solution

2x−4 2x−4

Exercise 86

Add 4( x 2 −2x−3) 4( x 2 −2x−3) to −6( x 2 −5) −6( x 2 −5) .

Exercise 87

Subtract 3 times (2x−1) (2x−1) from 8 times (x−4) (x−4) .

Solution

2x−29 2x−29

Exercise 88

(x+4)(x−6) (x+4)(x−6)

Exercise 89

(x−3)(x−8) (x−3)(x−8)

Solution

x 2 −11x+24 x 2 −11x+24

Exercise 90

(2a−5)(5a−1) (2a−5)(5a−1)

Exercise 91

(8b+2c)(2b−c) (8b+2c)(2b−c)

Solution

16 b 2 −4bc−2 c 2 16 b 2 −4bc−2 c 2

Exercise 92

(a−3) 2 (a−3) 2

Exercise 93

(3−a) 2 (3−a) 2

Solution

a 2 −6a+9 a 2 −6a+9

Exercise 94

(x−y) 2 (x−y) 2

Exercise 95

(6x−4) 2 (6x−4) 2

Solution

36 x 2 −48x+16 36 x 2 −48x+16

Exercise 96

(3a−5b) 2 (3a−5b) 2

Exercise 97

(−x−y) 2 (−x−y) 2

Solution

x 2 +2xy+ y 2 x 2 +2xy+ y 2

Exercise 98

(k+6)(k−6) (k+6)(k−6)

Exercise 99

(m+1)(m−1) (m+1)(m−1)

Solution

m 2 −1 m 2 −1

Exercise 100

(a−2)(a+2) (a−2)(a+2)

Exercise 101

(3c+10)(3c−10) (3c+10)(3c−10)

Solution

9 c 2 −100 9 c 2 −100

Exercise 102

(4a+3b)(4a−3b) (4a+3b)(4a−3b)

Exercise 103

(5+2b)(5−2b) (5+2b)(5−2b)

Solution

25−4 b 2 25−4 b 2

Exercise 104

(2y+5)(4y+5) (2y+5)(4y+5)

Exercise 105

(y+3a)(2y+a) (y+3a)(2y+a)

Solution

2 y 2 +7ay+3 a 2 2 y 2 +7ay+3 a 2

Exercise 106

(6+a)(6−3a) (6+a)(6−3a)

Exercise 107

( x 2 +2)( x 2 −3) ( x 2 +2)( x 2 −3)

Solution

x 4 − x 2 −6 x 4 − x 2 −6

Exercise 108

6(a−3)(a+8) 6(a−3)(a+8)

Exercise 109

8(2y−4)(3y+8) 8(2y−4)(3y+8)

Solution

48 y 2 +32y−256 48 y 2 +32y−256

Exercise 110

x(x−7)(x+4) x(x−7)(x+4)

Exercise 111

m 2 n(m+n)(m+2n) m 2 n(m+n)(m+2n)

Solution

m 4 n+3 m 3 n 2 +2 m 2 n 3 m 4 n+3 m 3 n 2 +2 m 2 n 3

Exercise 112

(b+2)( b 2 −2b+3) (b+2)( b 2 −2b+3)

Exercise 113

3p( p 2 +5p+4)( p 2 +2p+7) 3p( p 2 +5p+4)( p 2 +2p+7)

Solution

3 p 5 +21 p 4 +63 p 3 +129 p 2 +84p 3 p 5 +21 p 4 +63 p 3 +129 p 2 +84p

Exercise 114

(a+6) 2 (a+6) 2

Exercise 115

(x−2) 2 (x−2) 2

Solution

x 2 −4x+4 x 2 −4x+4

Exercise 116

(2x−3) 2 (2x−3) 2

Exercise 117

( x 2 +y) 2 ( x 2 +y) 2

Solution

x 4 +2 x 2 y+ y 2 x 4 +2 x 2 y+ y 2

Exercise 118

(2m-5n) 2 (2m-5n) 2

Exercise 119

(3 x 2 y 3 −4 x 4 y) 2 (3 x 2 y 3 −4 x 4 y) 2

Solution

9 x 4 y 6 −24 x 6 y 4 +16 x 8 y 2 9 x 4 y 6 −24 x 6 y 4 +16 x 8 y 2

Exercise 120

(a−2) 4 (a−2) 4

Terminology Associated with Equations ((Reference))

Find the domain of the equations for the following problems.

Exercise 121

y=8x+7 y=8x+7

Solution

all real numbers

Exercise 122

y=5 x 2 −2x+6 y=5 x 2 −2x+6

Exercise 123

y= 4 x−2 y= 4 x−2

Solution

all real numbers except 2

Exercise 124

m= −2x h m= −2x h

Exercise 125

z= 4x+5 y+10 z= 4x+5 y+10

Solution

x can equal any real number; y can equal any number except−10 x can equal any real number; y can equal any number except−10

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