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1
1.1
FUNDAMENTALS OF ALGEBRA
Real Numbers
Concept Questions
page 6
1. a. 4 (answer is not unique).
c.
2.
3
4
1
2
b. 0
d.
(answer is not unique).
b.
0 5
1
3
4. a. No. For example, 4
4
5 2
2
5
4
b. No.
5. If ab
2
5
ac
4
a
3
b
c
b
c .
7
a
4
b
2
3 1415
c.
5
2
5
3, and
8
5
425
5
2
5
.
0, then neither a nor b is equal to zero. If abc
Exercises
c. π
0 3333
3. a. The associative law of addition states that a
b. The distributive law states that ab
3 (answer is not unique).
0, then none of a, b, and c is equal to zero.
page 6
1. The number
3 is an integer, a rational number, and a real number.
2. The number
420 is an integer, a rational number, and a real number.
3. The number38 i
sarationalrealnumber.
4. The number
5. The number
11 is an irrational real number.
6. The number
7. The number
2 is
8. The number2
9. The number 2
11. False.
an irrational real number.
421 is a rational real number.
4
125
sarationalrealnumber.
i
5 is an irrational real number.
i
sanirrationalrealnumber.
10. The number 2 71828
number.
is an irrational real
2 is not a whole number.
12. True.
13. True.
14. True.
15. False. No natural number is irrational.
16. True.
17.
2x
y
z
z
2x
y : The Commutative Law of Addition.
1
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2
1 FUNDAMENTALS OF ALGEBRA
18. 3x
z
2y
19. u
3x
3
20. a2
u: The Commutative Law of Multiplication.
3
b2c
21. u
z: The Associative Law of Addition.
2y
a2b2
c: The Associative Law of Multiplication.
2
2u
u : The Distributive Law.
22.
2u
2u
: The Distributive Law.
23.
2x
3y
24.
a
2b
x
a
25. a
[
c
26.
2x
y
27. 0
2a
28. If
x
29. If
x
30. If x
4y
a
3b
d ]
x 4y
3y
a
a
3x
3b
3b
c
: The Associative Law of Addition.
a
2b
3b : The Distributive Law.
d : Property 1 of negatives.
2y
y
2x
3x
2y : Property 3 of negatives.
0: Property 1 involving zero.
y
x
2
y
2x
2x
2x
0, then x
0, then x
5
0, then x
9
y or x
y. Property 2 involving zero.
52
2, or x
0 or x
9
. Property 2 involving zero.
Property 2 involving zero.
2.
31.
x
x
1
3
2x
1
x
3
2x
1
x
3
2x
1
x
3
x
2x
2
x
32.
33.
a b
b
34.
x 2y
3x
y
35.
36.
a
6x
.Property2ofquotients.
1
2x
1
.Property2ofquotients.
a
2y
x
3x
a
2
2y
y
b
. Properties 2 and 5 of quotients.
a b
y
3x
2 x
x
2y
x
. Properties 2 and 5 of quotients and the Distributive Law.
2
b bc c
a b
. Property 6 of quotients and the Distributive Law.
b c
b
x y
x 1
1
b
b
b
a
a
ab
b
a b
a
b c
1
y
x
x x
37. False. Consider a
1
.Property7ofquotientsandtheDistributiveLaw.
12
2 and b
. Then ab
1, but a
1 and b
1.
1
38. True. Multiplying both sides of the equation by a (whichexistsbecausea
39. False. Consider a
40. False. Consider a
3 and b
2. Then a
3 and b
2. Then
b
3
a
b
2
b
3
2
b
a
3
a
.
2
2
3
1.
1
ab
0),wehavea
0.
a
0
,orb
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3
1.2 POLYNOMIALS
41. False. Consider a
1, b
42. False. Consider a
1.2
2, and c
1, b
a
3. Then
2, and c
b
c
1
a
b
c
3. Then
2
3
a
4
b
1
3
a b
1 2
1
2 3
2
c
3
6
c
1
2
3
2.
.
Polynomials
Concept Questions
page 12
1. a. No, this is not a polynomial expression because of the term of 2
x in which the power of x is not a nonnegative
integer.
b. Yes
c. No. It is a rational expression.
2. a. A polynomial of degree n in x is an expression of the form an xn an 1 xn 1
nonnegative integer and a0 a1
an are real numbers with an
is
x4 2x3 2x2 5x 7.
b. One polynomial of degree 3 in x and y is 2x
3. (a) 1
b2
2b
Exercises
1. 34
3.
5.
6x y2
b2
2ab
c. a2
3
3
3
2.
81.
5
2
2
2
8
3
34
9. 23
11.
13.
3y
2x
3
3
3
3
3
3
5
3
2
3
4.
27
2
3
25
28
2
3y
3
3
5
3
5
81
125 .
3
4x
2x
5
16. 3x2
5xy
2y
17. 5y2
2y
1
2x2
3x
4
3y
6
5
243y5.
2x
2x2
4
y2
4x
5x
2x
6
5y2
8
6
6x
7x2
4
2x2
3xy
4y
x2
3
2x2
3.
2x
3x2
2x2
3
2
2
2
2y
1
3x
4
x2
3
3
12.
2x
3
14.
3x
2
5x
3xy
4y
2x
2y
5y2
8
6
3x2
4
5
5
3
2x
2
2x
y2
2x2
5x
10.
4y
243.
5
32x5.
3x
2x
2xy
2y
125 .
3
16 .
5
3
x2
64
27
64
3
7x2
4
4
3
4x
4
16 .
4
9
4
2
3
5 2x2
y2
4
5
2
3
2
9
4
5
10.
5xy
3
4
4
8.
256.
3
4
6.
81.
3
5
15. 7x2
18.
2
2
3
3
b2
32.
3
3
7.
6y3 (answer is not unique).
page 12
3
2
b. a2
4x2 y
xy
3y
a1 x a0, where n is a
0. One polynomial of degree 4 in x
2
5x
2y
4.
1
8
4x
5 4
4y2
3
7x
9x2
2y
5.
3x
9.
1.
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4
1 FUNDAMENTALS OF ALGEBRA
2 4x3 3x2 1 7x 6 2
8x 2
19.
1 2x3
1 2x2
0 8x
2 4x3
2
1 2x3
20. 1 4x3
1 2x2
3 2
0 8x3
2 1x
1 7x
4 2x2
1 4x3
1 8
3x2
2 5x
1 2x2
2 2x
21. 3x2
6x5.
2r s2
23.
2x x2
2
24. x y
2y
3x
25. 2m
3m
4
4r2s2
4x3
2x3
2x y2
3x2 y.
m
3x
5
b
27. 3
2a
b
4
28. 2
3m
1
3
2
2x
30.
3r
1
2r
5
3r
31.
2x
3y
3x
32.
5m
2n
5m
33.
3r
34.
2m
35.
0 2x
5m
3m
1 2y
3r
4r
2m
0 3x
2 1y
2x3
6x
3b
4b
6n
18m
6n
2.
2
6x2
4x
9x
6r2
5
3y
3x
5m
4r
3s
2n
3n
3m
0 2x
0 3x
2r
6x2
6r2
4x y
12r2
2n
5x
13r
6.
5.
6y2
9x y
25m2
3n
9x.
7b.
6
5
6x2
2y
2n
2s
15r
14a
9x2
15mn
6x2
6n2
10mn
9r s
8r s
6s2
12r2
6m2
4mn
9mn
6n2
2 1y
1 2y
0 3x
6y2.
5x y
25m2
rs
6m2
6n2.
6s2.
5mn
0 06x2
2 1y
5mn
6n2.
0 42x y
2 52y2
3 2m
1 7n
4 2m
1 3n
y
3x2
38. 3m
2n2
39.
2x
3y
2
40.
3m
2n
2
41.
2u
2x 3x2
2y
2m2
3 2m
2x
3m
2u
2
2
2
2x
2
2u
3m
2
4 2m
3n
3y
2
2n2
2n
4u2
6x3
2m2
2
2
.
4x2
2
1 7n
9m2
3x2y
4xy
3n
6m3
12x y
9y2.
12mn
4 2m
2 21n2
7 14mn
2y
3y
2n
1 3n
4 16mn
y 3x2
2y
3m 2m2
3n
2 52y2.
0 06x y
13 44m2
2x
15x
12m
3n
3m
9x2
3x
3s
2n
9m.
8a
2y
5m
7m2
6a
2r
3x
m
8a
3
5
2x
3s
2
2
2r
3n
4b
6m
3x
2y
6x3
0 06x2
37.
4x3
5.
4x.
m2
8m
3b
2n
3x
2x3
3
6a
3
3n
x2
2a
4m
4r
6m2
2x
2x
2s
4x3
4x
1
29.
36.
2 1x
1 2x
16r3s5.
2s
m
2x2
3x
0 36x y
8
2
2x3
22.
26.
0
2 1x
1
3
1 2x2
8 2.
0 8x3
3 2
1 2x3
6 2
4n2.
9mn
1 3n
13 44m2
2 98mn
2y2.
4m2n2
6n3.
2 21n2.
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5
1.2 POLYNOMIALS
42.
3r
4s
43.
2x
1
44.
3m
2
2
45.
2x
3y
2
2
3r
4s
3r
3x
2 x2
1
2m
2
2y
9m2
4
1
3x
9r2
4x2
3
m
1
2
4s
2
16s2.
4x
1
3x
2x2
12m
4
2m
2m2
x
2
4x2
y
4x2
46.
x
t2
47.
y
2y
2t
2x y 3
3x
2t2
4
x
y
t2
1
3m2
49. 2x
2m2
1
3x
50. 3m
[x
m
2
3m
4
2x
m
[ x
2x
9t2
2t
4.
3m2
2m2
3m
4
6m4
9m3
2x
3x
[x
2x
4x
1
5 ]
52. 3x2
x2
x [x
1
2
2
x
3x
53.
54.
2x
x
3
2y
2
x ]
1
2
4
x
2
x
1
x
3
2x
4
y
3m
x
4x
1
1 ]
2
3x2
1
4
x
2
3y
x
x2
3x [2x
2x
56.
3
3x
x
2y
3
x ]
3x
3
2
x
2x2
3x
2y
x
2
3x
x
4
x
2x
2x
2y
2x
3
2x
3m
1
m
3m
15
4
1
x ]
3x
1.
x
2
2x
2y
3x
3y 3
3x2
7x y 2y2
4t3
8t2
t2
6m4
4
9m3
2t
3y 3.
4
12m2
2m2
3m
4
4x2
12x
4]
2
x
2m
2x
3x
x
m
5
4m
22
2x
2 [m
11
3
3m
x
1 ]
1
x
2
3m
2x
x
2
2x
5
2
1 ]
1
1.
[2x
x
1
x
2x
x
1
2x
4]
7m
2x
22.
1
2
2
2
2x2
3x
32
x
1
2
x2
x
1.
8
1
3 x2
16
2x
x2
10x
50.
3y2
2x2
3xy
2x
5x2
5xy 2y2
2x
3
9
3x2
48
2x
2 x2
3xy
xy
3xy 2x
7
2
2y
2x
4y
2.
2t4
3m
1
2
2x2
3x
2y
[3x
4x
2m
9x2
y
2m2
3
3
2x
1
2 [m
3y
6x y
x
4
2x
x2
2
10m.
6x y 2x y 3x
1]
2
9y2
2.
4.
2x
x
4xy 4y2
1
3m
2x
m
2
[2x
2
55. 2x
2x
14m2
x
x
x
2t
4t3
3m
51. x
t2
2t2
4
1 ]
3 [2m
2y2
x
11m2
4
9y2
6xy
2x2
3
12x y
x y 3x2
2t
2t4
48.
1
2
2
4xy 6y2
3x
2x
9x
2x2
2x2
x
y
3
3
2x2
x
3
3x
3
2x
3 x2
4xy
x2
4x2
4xy
16x y
2x
11x2
4y2
4y2
y2
3
10x
9x2
2x.
12xy
9x2
12x2
22x3
3
4y2
12xy
48x y
4y2
3y2.
20x2
6x.
4x2
y2
4x2
y2
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6
1 FUNDAMENTALS OF ALGEBRA
57. The total weekly profit is given by the revenue minus the cost:
0 04x2
0 000002x3
2000x
0 02x2
1000x
120,000
0 04x2
0 000002x3
58. The total revenue is given by x p x
0 0004x 10
2
the revenue minus the cost: 0 0004x
10x
0 0001x2
0 2t2
59. The total revenue is given by
where 0 t 12.
0 5t2
150t
0 000002x3
2000x
0 02x2
0 02x2
1000x
1000x
120,000
120,000.
0 0004x2
4x 400
10x. Therefore, the total profit is given by
0 0005x2 6x 400.
0 7t2
350t thousand dollars t months from now,
200t
60. In month t, the revenue of the second gas station will exceed that of the first gas station by
0 5t2
0 2t2
200t
0 3t2
150t
0 5t2
61. The difference is given by 12
3t
t
50t thousand dollars, where 0
54
0 75t
38 5
436 2
24 3t
365
12.
12 0 5t2
2 25t
15 5
6t2
27t
x3
x
186 dollars year.
62. The gap is given by 3 5t2
63. False. Let a
2, b
3, m
26 7t
2. Then 23
3, and n
32
8
9
3 5t2
72
2
2 4t
3 2
3
71 2.
65.
64. True.
65. False. For example, x2 1 is a polynomial of degree 2 and x is a polynomial of degree 1, but
is a polynomial of degree 3, not 2.
x3
66. False. For example, p
p q x3 x 1
1.3
1 is a polynomial of degree 3 and q
x3
2
x 3 is a polynomial of degree 1.
x
x3
x2
1
x
2 is a polynomial of degree 3, but
Factoring Polynomials
Concept Questions
page 18
1. A polynomial is completely factored over the set of integers if it is expressed as a product of prime polynomials
with integral coefficients. An example is 4x2
2. a.
a
Exercises
1. 6m2
3. 9ab2
a2
b
9y2
2x
b2
ab
3y
b.
2x
a
3y .
b
a2
ab
4t3
t
b2
page 18
4m
2m
6a2b
3m
3ab
5. 10m2n
15mn2
7. 3x 2x
1
3b
20mn
5 2x
2. 4t4
2 .
1
4. 12x3 y5
2a .
5mn
2m
2x
12t3
1
3n
3x
4 .
5 .
6.
6x4y
8. 2u 3
16x2 y3
4x2y2
2
3 .
4x2 y3
2x2y3
5
3
2
3x y2
4 .
2x2y 3x2
3
2y
2
y2 .
2u
5
.
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7
1.3 FACTORING POLYNOMIALS
9.
3a
b
2
2u
10. 4u
11. 2m2
6
3x
17. 4a2
b2
19. u2
2
x
6u2
b
2a
b .
u
2
2
u
2
2ab
21. z2
4 is prime.
23. x2
6x y
25. x2
3x
2
12. 6x2
x
14. 2u2
5u
16. m2
2m
x
x
18m
3a
1
3x
2
3u .
1
2
12
2x
2u
b
4ac
u
4
2x
y
2ad .
1 .
3
.
3 is prime.
3 4x2
y2
3
2x
y .
2ab
5c
2ab
5c .
2
25
is prime.
12u
9
2
2u
2
3
.
1 .
m2
3m
2u
3y2
24. 4u2
4
d
2c
.
y2 is prime.
3m2
2u
22. u2
4
d ]
2c
18. 12x2
u
5c
2a
2u
2y .
2a
25c2
26. 3m3
b
[3a
6 .
x
3y
2
4u
m
1
d
2c
1 is prime.
2
20. 4a2b2
2
d
2c
2u
2m
6y2
xy
15. x2
2a
6u2
11m
13. x2
d
2c
m
6
m
3m
m
3
2
.
27. 12x2 y
10x y
28. 12x2 y
2x y
29. 35r2
r
2
30. 6u
12y
24y
12
6
31. 9x3 y
4x y3
xy
32. 4u4
9u2
35.
u2
a
36. 2x
3
2b
x
y
2
5x
6
2y
3x
2
2x
3 .
6x2
x
12
2y
3x
4
2x
3 .
5r
3 .
2
xy
3x
3u
4y2
4u2
2
2
2
9
2 .
xy
9
4y
16u4
2
6x2
2u
9x2
x2
9
4
3
3
16y2
34. 16u4
2y
7r
9u
33. x4
2y
u2
x2
4u2
2b
2
[ a
8x
x
y2
2
2b
2x
2x y
37.8m3
1
2m
3
1
2m
2u
4y
2
a
1
2
3x
x
y
2y
2
2
3
x2
4y .
2
3
2
a
2b ] [ a
2
4 x
y2
y
x
2y2
3x
4m2
2m
1 .
u2
2u
4u2
3
2x
2y2 .
x
3x
3
2b ]
y
2y .
2u
4u2
a
2b
2
2y
2 x
3
3
.
.
4b
y2
2a
8ab.
x
y
2 x
y2
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8
1 FUNDAMENTALS OF ALGEBRA
38. 27m3
8
3m
39. 8r3
27s3
2r
40. x3
64y3
41. u2
6
42. r6s6
3
3
x3
8u2
u2
8s3
s3
6ay
bx
2by
46. 6ux
4uy
3
49. 4x3
9x y2
50. 4u4
51. x4
3x3
52. a2
b2
a
a
c
54. ax2
1
ab
y
2u
3x
2 2
u
2
x
3y
2
u2
b
a
au2
c
by2
xy
b
4 .
1 .
u2
2y
b .
3a
3x
u
2 .
2y
2u
.
u2
u
2
.
.
y 4x2
2x
x
b
au
au
cu
c
ax2
xy
abx y
u
by2
ax
2
3
2
u2
2u
x3
x
y
.
2 .
b
1
3y
y .
3
a
2
2x
2u
3
a
9y2
x
3y
2
3
x
2r2s
1
2y
2
9y2
2x
2
x
2y
2
2
2x
2 u2
2
u2
r4s2
2
3
3x
4x2
3
u2
2y
u2
4u2
x2
x
2
x
x3
b
u2
3
4
a
53. au2
2y
4 .
s3 r2s
3
2 u4
4
x
9y3
b
x2
3
2
2
23
3
9s2 .
16y2 .
4
2
r2s
2u2
u2
6
4x y
3a
2
4
2x
x2
x2
2
3
6r s
s3
2u4
x
2
4r2
8
2x
4x2 y
11u2
3s
2
45. 3ax
6
2r
u2
4
2
3
8
2u2
u2
4 .
6
4u2
48. u4
6m
4y
44. 2u4
u2
9m2
x
r6s3
3
2
3
6x
4
3s
3m
4y
43. 2x3
47. u4
x2
23
a
b
c
u
x
by
56.
t3
1 .
u
1
y
x
au
1
by
c .
x
by
ax
y
.
55. P
Pr t
57. 8000x
59. k M x
61. R
P
1
100x2
kx2
60,000
kx
100x
rt .
x .
100x
80
M
x .
x2
200
6t2
15t
t
t2
6t
k Qx
kx2
kx
Q
x .
0 1x2
500x
58. R
60.
x
300
x .
1
62. T
2
1
2t
63. V
V0
V0
0
T
273 2V73
273
T
.
64.
kD2
2
0 1x
x
1
t3
39t2
360t
t
15
t
D3
3
D2
k
2
5000 .
2t
24 .
3 .
15 .
t2
39t
360
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9
1.4 RATIONAL EXPRESSIONS
1.4
Rational Expressions
Concept Questions
page 25
1. a. Quotients of polynomials are rational expressions;
2x2
2
3x
1
.
4
3x
P
b. Any polynomial P can be written in the form
1,
butnotallrationalexpressionscanbewrittenasapolynomial.
PR
2. a. QS ; PS
RQ
Exercises
1.
28x
7x3
3.
4x
5x
4
2.
x.
4 x
12
15
9.
x
x2
6x
12.
8r3 s3
2r2 r s s2
17.
y2
xy
1
3x2
2
15
xy
x2
xy
y2
2r s
2r
s
r
2x8
16x5
8x2
15x4
5x6
10y
4
2x2
7x
10.
.
1
4r2
6
6
2m
1
2
18m
9
9
2m
1
3
2
4y
22
y
4y
2
62
y
3
y
1
11y
4y
3
12
4r
2
2
16.
2
5x
5x
18.
25y4
3y2
12y
5y3
6x5
4x
6x5
7x3
21x2
7x3
21x2
4x
4y
12
x
6
x2
x
6
y
2
y
2
2y 3
.
2y 1
3y 1
2y 3
2y 3
.
2y 3
4
y 2
3y
3
3
3
.
4 2
y .
y
6
20. 4
y
5
3y 6
2y 1
2y
3r 2
2r 1 6 r 2
2 r 2 2 2r 1
2x2
y
3y 1
2y 3
9
2y
2
y 1
y 1
2y 3
2y 1
8
6 y
24
12
3
2 6r
4y y
8y
y
2
2
r s
6
2 m
9
8y2
8y
3
.
s2
2r s
2.
6 x
6
6
3
s
3 m
x
12
m
2y
s2
3x 5 x 2y
3
x
y .
x y.
4r2
16x5
r
8.
14.
6
6r2
16 2
18y2
y2
s
3x3
5x
2r
22.
1
.
x 1
x.
3m
2
6.
x
2x
x2
6
21.
x
y
6
19.
x 1
x 1
x4
3x
x 2y
2m
1
2x
2r
8
3x3
8x2
2
x
x2
2x
x 3
x 3
2x 1
x 3
3
11.
15.
.
1
x 2
x 2
2
2
5x
6x3
32
4.
3y4
2
x3 y 3
13.
5
2x
9
2x2
4
3
3x
x
3x
x2
3
5 x
2
5. 6x 2 3x
6x
7.
Q
R
;P
R
page 25
2
2
P Q
b.
x 3
x 2
2x 3
2x 3
x 2
x
3
3r
2
2
x 2
3
x
2
x
3
.
x 3
4 y
y
3 y
2
2y
1
5
2x .
3 3 y
2
2
1
2y
12 y
3
2y 1
2
3
4 2y 3
6 y 4
2
y .
y 4