College Algebra 2e

# Chapter 1

### 1.1Real Numbers: Algebra Essentials

1.
1. $11 1 11 1$
2. $3 1 3 1$
3. $− 4 1 − 4 1$
2.
1. 4 (or 4.0), terminating;
2. $0. 615384 ¯ , 0. 615384 ¯ ,$ repeating;
3. –0.85, terminating
3.
1. rational and repeating;
2. rational and terminating;
3. irrational;
4. rational and terminating;
5. irrational
4.
1. positive, irrational; right
2. negative, rational; left
3. positive, rational; right
4. negative, irrational; left
5. positive, rational; right
5.
N W I Q Q'
a. $− 35 7 − 35 7$ X X
b. 0 X X X
c. $169 169$ X X X X
d. $24 24$ X
e. 4.763763763... X
6.
1. 10
2. 2
3. 4.5
4. 25
5. 26
7.
1. 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
2. 33, distributive property;
3. 26, distributive property;
4. $4 9 , 4 9 ,$ commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
5. 0, distributive property, inverse property of addition, identity property of addition
8.
Constants Variables
a. $2πr( r+h ) 2πr( r+h )$ $2,π 2,π$ $r,h r,h$
b. 2(L + W) 2 L, W
c. $4 y 3 +y 4 y 3 +y$ 4 $y y$
9.
1. 5;
2. 11;
3. 9;
4. 26
10.
1. 4;
2. 11;
3. $121 3 π 121 3 π$ ;
4. 1728;
5. 3
11.

1,152 cm2

12.
1. $2 t −1; 2 t −1;$
2. $3pq−4p+q; 3pq−4p+q;$
3. $7r−2s+6 7r−2s+6$
13.

$A=P( 1+rt ) A=P( 1+rt )$

### 1.2Exponents and Scientific Notation

1.
1. $k 15 k 15$
2. $( 2 y ) 5 ( 2 y ) 5$
3. $t 14 t 14$
2.
1. $s 7 s 7$
2. $( −3 ) 5 ( −3 ) 5$
3. $( e f 2 ) 2 ( e f 2 ) 2$
3.
1. $( 3y ) 24 ( 3y ) 24$
2. $t 35 t 35$
3. $( −g ) 16 ( −g ) 16$
4.
1. $1 1$
2. $1 2 1 2$
3. $1 1$
4. $1 1$
5.
1. $1 ( −3t ) 6 1 ( −3t ) 6$
2. $1 f 3 1 f 3$
3. $2 5 k 3 2 5 k 3$
6.
1. $t −5 = 1 t 5 t −5 = 1 t 5$
2. $1 25 1 25$
7.
1. $g 10 h 15 g 10 h 15$
2. $125 t 3 125 t 3$
3. $−27 y 15 −27 y 15$
4. $1 a 18 b 21 1 a 18 b 21$
5. $r 12 s 8 r 12 s 8$
8.
1. $b 15 c 3 b 15 c 3$
2. $625 u 32 625 u 32$
3. $−1 w 105 −1 w 105$
4. $q 24 p 32 q 24 p 32$
5. $1 c 20 d 12 1 c 20 d 12$
9.
1. $v 6 8 u 3 v 6 8 u 3$
2. $1 x 3 1 x 3$
3. $e 4 f 4 e 4 f 4$
4. $27r s 27r s$
5. $1 1$
6. $16 h 10 49 16 h 10 49$
10.
1. $1.52× 10 5 1.52× 10 5$
2. $7.158× 10 9 7.158× 10 9$
3. $8.55× 10 13 8.55× 10 13$
4. $3.34× 10 −9 3.34× 10 −9$
5. $7.15× 10 −8 7.15× 10 −8$
11.
1. $703,000 703,000$
2. $−816,000,000,000 −816,000,000,000$
3. $−0.00000000000039 −0.00000000000039$
4. $0.000008 0.000008$
12.
1. $−8.475× 10 6 −8.475× 10 6$
2. $8× 10 −8 8× 10 −8$
3. $2.976× 10 13 2.976× 10 13$
4. $−4.3× 10 6 −4.3× 10 6$
5. $≈1.24× 10 15 ≈1.24× 10 15$
13.

Number of cells: $3× 10 13 ; 3× 10 13 ;$ length of a cell: $8× 10 −6 8× 10 −6$ m; total length: $2.4× 10 8 2.4× 10 8$ m or $240,000,000 240,000,000$ m.

1.
1. $15 15$
2. $3 3$
3. $4 4$
4. $17 17$
2.

$5| x || y | 2yz . 5| x || y | 2yz .$ Notice the absolute value signs around x and y? That’s because their value must be positive!

3.

$10| x | 10| x |$

4.

$x 2 3 y 2 . x 2 3 y 2 .$ We do not need the absolute value signs for $y 2 y 2$ because that term will always be nonnegative.

5.

$b 4 3ab b 4 3ab$

6.

$13 5 13 5$

7.

$0 0$

8.

$6 6 6 6$

9.

$14−7 3 14−7 3$

10.
1. $−6 −6$
2. $6 6$
3. $88 9 3 88 9 3$
11.

$( 9 ) 5 = 3 5 =243 ( 9 ) 5 = 3 5 =243$

12.

$x (5y) 9 2 x (5y) 9 2$

13.

$28 x 23 15 28 x 23 15$

### 1.4Polynomials

1.

The degree is 6, the leading term is $− x 6 , − x 6 ,$ and the leading coefficient is $−1. −1.$

2.

$2 x 3 +7 x 2 −4x−3 2 x 3 +7 x 2 −4x−3$

3.

$−11 x 3 − x 2 +7x−9 −11 x 3 − x 2 +7x−9$

4.

$3 x 4 −10 x 3 −8 x 2 +21x+14 3 x 4 −10 x 3 −8 x 2 +21x+14$

5.

$3 x 2 +16x−35 3 x 2 +16x−35$

6.

$16 x 2 −8x+1 16 x 2 −8x+1$

7.

$4 x 2 −49 4 x 2 −49$

8.

$6 x 2 +21xy−29x−7y+9 6 x 2 +21xy−29x−7y+9$

### 1.5Factoring Polynomials

1.

$( b 2 −a)(x+6) ( b 2 −a)(x+6)$

2.

$(x−6)(x−1) (x−6)(x−1)$

3.
1. $(2x+3)(x+3) (2x+3)(x+3)$
2. $( 3x−1 )( 2x+1 ) ( 3x−1 )( 2x+1 )$
4.

$(7x−1) 2 (7x−1) 2$

5.

$(9y+10)(9y−10) (9y+10)(9y−10)$

6.

$(6a+b)(36 a 2 −6ab+ b 2 ) (6a+b)(36 a 2 −6ab+ b 2 )$

7.

$(10x−1)( 100 x 2 +10x+1 ) (10x−1)( 100 x 2 +10x+1 )$

8.

$(5a−1) − 1 4 (17a−2) (5a−1) − 1 4 (17a−2)$

### 1.6Rational Expressions

1.

$1 x+6 1 x+6$

2.

$(x+5)(x+6) (x+2)(x+4) (x+5)(x+6) (x+2)(x+4)$

3.

$1 1$

4.

$2(x−7) (x+5)(x−3) 2(x−7) (x+5)(x−3)$

5.

$x 2 − y 2 x y 2 x 2 − y 2 x y 2$

### 1.1 Section Exercises

1.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

3.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

5.

$−6 −6$

7.

$−2 −2$

9.

$−9 −9$

11.

9

13.

-2

15.

4

17.

0

19.

9

21.

25

23.

$−6 −6$

25.

17

27.

4

29.

$14 14$

31.

$−66 −66$

33.

$–12 –12$

35.

$–44 –44$

37.

$–2 –2$

39.

$−14y−11 −14y−11$

41.

$−4b+1 −4b+1$

43.

$43z−3 43z−3$

45.

$9y+45 9y+45$

47.

$−6b+6 −6b+6$

49.

$16x 3 16x 3$

51.

$9x 9x$

53.

$1 2 ( 40−10 )+5 1 2 ( 40−10 )+5$

55.

irrational number

57.

$g+400−2( 600 )=1200 g+400−2( 600 )=1200$

59.

61.

68.4

63.

true

65.

irrational

67.

rational

### 1.2 Section Exercises

1.

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So $2 3 2 3$ is the same as $2×2×2, 2×2×2,$ which is 8. $3 2 3 2$ is the same as $3×3, 3×3,$ which is 9.

3.

It is a method of writing very small and very large numbers.

5.

81

7.

243

9.

$1 16 1 16$

11.

$1 11 1 11$

13.

1

15.

$4 9 4 9$

17.

$12 40 12 40$

19.

$1 7 9 1 7 9$

21.

$3.14× 10 −5 3.14× 10 −5$

23.

16,000,000,000

25.

$a 4 a 4$

27.

$b 6 c 8 b 6 c 8$

29.

$a b 2 d 3 a b 2 d 3$

31.

$m 4 m 4$

33.

$q 5 p 6 q 5 p 6$

35.

$y 21 x 14 y 21 x 14$

37.

$25 25$

39.

$72 a 2 72 a 2$

41.

$c 3 b 9 c 3 b 9$

43.

$y 81 z 6 y 81 z 6$

45.

0.00135 m

47.

$1.0995× 10 12 1.0995× 10 12$

49.

0.00000000003397 in.

51.

12,230,590,464 $m 66 m 66$

53.

$a 14 1296 a 14 1296$

55.

$n a 9 c n a 9 c$

57.

$1 a 6 b 6 c 6 1 a 6 b 6 c 6$

59.

0.000000000000000000000000000000000662606957

### 1.3 Section Exercises

1.

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

3.

The principal square root is the nonnegative root of the number.

5.

16

7.

10

9.

14

11.

$7 2 7 2$

13.

$9 5 5 9 5 5$

15.

25

17.

$2 2$

19.

$2 6 2 6$

21.

$5 6 5 6$

23.

$6 35 6 35$

25.

$2 15 2 15$

27.

$6 10 19 6 10 19$

29.

$− 1+ 17 2 − 1+ 17 2$

31.

$7 2 3 7 2 3$

33.

$15 5 15 5$

35.

$20 x 2 20 x 2$

37.

$7 p 7 p$

39.

$17 m 2 m 17 m 2 m$

41.

$2b a 2b a$

43.

$15x 7 15x 7$

45.

$5 y 4 2 5 y 4 2$

47.

$4 7d 7d 4 7d 7d$

49.

$2 2 +2 6x 1−3x 2 2 +2 6x 1−3x$

51.

$−w 2w −w 2w$

53.

$3 x − 3x 2 3 x − 3x 2$

55.

$5 n 5 5 5 n 5 5$

57.

$9 m 19m 9 m 19m$

59.

$2 3d 2 3d$

61.

$3 2 x 2 4 2 3 2 x 2 4 2$

63.

$6z 2 3 6z 2 3$

65.

500 feet

67.

$−5 2 −6 7 −5 2 −6 7$

69.

$mnc a 9 cmn mnc a 9 cmn$

71.

$2 x+1 2 2 2 x+1 2 2$

73.

$3 3 3 3$

### 1.4 Section Exercises

1.

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

3.

Use the distributive property, multiply, combine like terms, and simplify.

5.

2

7.

8

9.

2

11.

$4 x 2 +3x+19 4 x 2 +3x+19$

13.

$3 w 2 +30w+21 3 w 2 +30w+21$

15.

$11 b 4 −9 b 3 +12 b 2 −7b+8 11 b 4 −9 b 3 +12 b 2 −7b+8$

17.

$24 x 2 −4x−8 24 x 2 −4x−8$

19.

$24 b 4 −48 b 2 +24 24 b 4 −48 b 2 +24$

21.

$99 v 2 −202v+99 99 v 2 −202v+99$

23.

$8 n 3 −4 n 2 +72n−36 8 n 3 −4 n 2 +72n−36$

25.

$9 y 2 −42y+49 9 y 2 −42y+49$

27.

$16 p 2 +72p+81 16 p 2 +72p+81$

29.

$9 y 2 −36y+36 9 y 2 −36y+36$

31.

$16 c 2 −1 16 c 2 −1$

33.

$225 n 2 −36 225 n 2 −36$

35.

$−16 m 2 +16 −16 m 2 +16$

37.

$121 q 2 −100 121 q 2 −100$

39.

$16 t 4 +4 t 3 −32 t 2 −t+7 16 t 4 +4 t 3 −32 t 2 −t+7$

41.

$y 3 −6 y 2 −y+18 y 3 −6 y 2 −y+18$

43.

$3 p 3 − p 2 −12p+10 3 p 3 − p 2 −12p+10$

45.

$a 2 − b 2 a 2 − b 2$

47.

$16 t 2 −40tu+25 u 2 16 t 2 −40tu+25 u 2$

49.

$4 t 2 + x 2 +4t−5tx−x 4 t 2 + x 2 +4t−5tx−x$

51.

$24 r 2 +22rd−7 d 2 24 r 2 +22rd−7 d 2$

53.

$32 x 2 −4x−3 32 x 2 −4x−3$ m2

55.

$32 t 3 −100 t 2 +40t+38 32 t 3 −100 t 2 +40t+38$

57.

$a 4 +4 a 3 c−16a c 3 −16 c 4 a 4 +4 a 3 c−16a c 3 −16 c 4$

### 1.5 Section Exercises

1.

The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, $4 x 2 4 x 2$ and $−9 y 2 −9 y 2$ don’t have a common factor, but the whole polynomial is still factorable: $4 x 2 −9 y 2 =( 2x+3y )( 2x−3y ). 4 x 2 −9 y 2 =( 2x+3y )( 2x−3y ).$

3.

Divide the $x x$ term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

5.

$7m 7m$

7.

$10 m 3 10 m 3$

9.

$y y$

11.

$( 2a−3 )( a+6 ) ( 2a−3 )( a+6 )$

13.

$( 3n−11 )( 2n+1 ) ( 3n−11 )( 2n+1 )$

15.

$( p+1 )( 2p−7 ) ( p+1 )( 2p−7 )$

17.

$( 5h+3 )( 2h−3 ) ( 5h+3 )( 2h−3 )$

19.

$( 9d−1 )( d−8 ) ( 9d−1 )( d−8 )$

21.

$( 12t+13 )( t−1 ) ( 12t+13 )( t−1 )$

23.

$(4x+10)(4x−10) (4x+10)(4x−10)$

25.

$(11p+13)(11p−13) (11p+13)(11p−13)$

27.

$(19d+9)(19d−9) (19d+9)(19d−9)$

29.

$(12b+5c)(12b−5c) (12b+5c)(12b−5c)$

31.

$( 7n+12 ) 2 ( 7n+12 ) 2$

33.

$( 15y+4 ) 2 ( 15y+4 ) 2$

35.

$(5p−12) 2 (5p−12) 2$

37.

$(x+6)( x 2 −6x+36) (x+6)( x 2 −6x+36)$

39.

$(5a+7)(25 a 2 −35a+49) (5a+7)(25 a 2 −35a+49)$

41.

$(4x−5)(16 x 2 +20x+25) (4x−5)(16 x 2 +20x+25)$

43.

$(5r+12s)(25 r 2 −60rs+144 s 2 ) (5r+12s)(25 r 2 −60rs+144 s 2 )$

45.

$( 2c+3 ) − 1 4 ( −7c−15 ) ( 2c+3 ) − 1 4 ( −7c−15 )$

47.

$( x+2 ) − 2 5 ( 19x+10 ) ( x+2 ) − 2 5 ( 19x+10 )$

49.

$( 2z−9 ) − 3 2 ( 27z−99 ) ( 2z−9 ) − 3 2 ( 27z−99 )$

51.

$( 14x−3 )( 7x+9 ) ( 14x−3 )( 7x+9 )$

53.

$( 3x+5 )( 3x−5 ) ( 3x+5 )( 3x−5 )$

55.

$(2x+5) 2 (2x−5) 2 (2x+5) 2 (2x−5) 2$

57.

$(4 z 2 +49 a 2 )(2z+7a)(2z−7a) (4 z 2 +49 a 2 )(2z+7a)(2z−7a)$

59.

$1 ( 4x+9 )( 4x−9 )( 2x+3 ) 1 ( 4x+9 )( 4x−9 )( 2x+3 )$

### 1.6 Section Exercises

1.

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

3.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

5.

$y+5 y+6 y+5 y+6$

7.

$3b+3 3b+3$

9.

$x+4 2x+2 x+4 2x+2$

11.

$a+3 a−3 a+3 a−3$

13.

$3n−8 7n−3 3n−8 7n−3$

15.

$c−6 c+6 c−6 c+6$

17.

$1 1$

19.

$d 2 −25 25 d 2 −1 d 2 −25 25 d 2 −1$

21.

$t+5 t+3 t+5 t+3$

23.

$6x−5 6x+5 6x−5 6x+5$

25.

$p+6 4p+3 p+6 4p+3$

27.

$2d+9 d+11 2d+9 d+11$

29.

$12b+5 3b−1 12b+5 3b−1$

31.

$4y−1 y+4 4y−1 y+4$

33.

$10x+4y xy 10x+4y xy$

35.

$9a−7 a 2 −2a−3 9a−7 a 2 −2a−3$

37.

$2 y 2 −y+9 y 2 −y−2 2 y 2 −y+9 y 2 −y−2$

39.

$5 z 2 +z+5 z 2 −z−2 5 z 2 +z+5 z 2 −z−2$

41.

$x+2xy+y x+xy+y+1 x+2xy+y x+xy+y+1$

43.

$2b+7a a b 2 2b+7a a b 2$

45.

$18+ab 4b 18+ab 4b$

47.

$a−b a−b$

49.

$3 c 2 +3c−2 2 c 2 +5c+2 3 c 2 +3c−2 2 c 2 +5c+2$

51.

$15x+7 x−1 15x+7 x−1$

53.

$x+9 x−9 x+9 x−9$

55.

$1 y+2 1 y+2$

57.

$4 4$

### Review Exercises

1.

$−5 −5$

3.

53

5.

$y=24 y=24$

7.

$32m 32m$

9.

whole

11.

irrational

13.

$16 16$

15.

$3 a 6 3 a 6$

17.

$x 3 32 y 3 x 3 32 y 3$

19.

$a a$

21.

$1.634× 10 7 1.634× 10 7$

23.

14

25.

$5 3 5 3$

27.

$4 2 5 4 2 5$

29.

$7 2 50 7 2 50$

31.

$10 3 10 3$

33.

$−3 −3$

35.

$3 x 3 +4 x 2 +6 3 x 3 +4 x 2 +6$

37.

$5 x 2 −x+3 5 x 2 −x+3$

39.

$k 2 −3k−18 k 2 −3k−18$

41.

$x 3 + x 2 +x+1 x 3 + x 2 +x+1$

43.

$3 a 2 +5ab−2 b 2 3 a 2 +5ab−2 b 2$

45.

$9p 9p$

47.

$4 a 2 4 a 2$

49.

$(4a−3)(2a+9) (4a−3)(2a+9)$

51.

$( x+5 ) 2 ( x+5 ) 2$

53.

$(2h−3k) 2 (2h−3k) 2$

55.

$(p+6)( p 2 −6p+36) (p+6)( p 2 −6p+36)$

57.

$(4q−3p)(16 q 2 +12pq+9 p 2 ) (4q−3p)(16 q 2 +12pq+9 p 2 )$

59.

$( p+3 ) 1 3 ( −5p−24 ) ( p+3 ) 1 3 ( −5p−24 )$

61.

$x+3 x−4 x+3 x−4$

63.

$1 2 1 2$

65.

$m+2 m−3 m+2 m−3$

67.

$6x+10y xy 6x+10y xy$

69.

$1 6 1 6$

### Practice Test

1.

rational

3.

$x=–2 x=–2$

5.

3,141,500

7.

$16 16$

9.

9

11.

$2x 2x$

13.

21

15.

$3 x 4 3 x 4$

17.

$21 6 21 6$

19.

$13 q 3 −4 q 2 −5q 13 q 3 −4 q 2 −5q$

21.

$n 3 −6 n 2 +12n−8 n 3 −6 n 2 +12n−8$

23.

$(4x+9)(4x−9) (4x+9)(4x−9)$

25.

$(3c−11)(9 c 2 +33c+121) (3c−11)(9 c 2 +33c+121)$

27.

$4z−3 2z−1 4z−3 2z−1$

29.

$3a+2b 3b 3a+2b 3b$

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