Finite Mathematics, Lial - Solutions, summaries, and outlines. 2022 updated

Chapter R


ALGEBRA REFERENCE
R.1 Polynomials 5. (0.613x 2 - 4.215x + 0.892) - 0.47(2 x 2 - 3x + 5)
Your Turn 1 = 0.613x 2 - 4.215x + 0.892 - 0.94 x 2
3( x 2 - 4 x - 5) - 4(3x 2 - 5x - 7) + 1.41x - 2.35

= 3x 2 - 12 x - 15 - 12 x 2 + 20 x + 28 = -0.327 x 2 - 2.805x - 1.458

= -9 x 2 + 8 x + 13 6. 0.5(5r 2 + 3.2r - 6) - (1.7r 2 - 2r - 1.5)
Your Turn 2 = (2.5r 2 + 1.6r - 3) + (-1.7r 2 + 2r + 1.5)
(3 y + 2)(4 y 2 - 2 y - 5) = (2.5r 2 - 1.7r 2 ) + (1.6r + 2r ) + (-3 + 1.5)
= (3 y )(4 y 2 - 2 y - 5) + (2)(4 y 2 - 2 y - 5) = 0.8r 2 + 3.6r - 1.5
= 12 y 3 - 6 y 2 - 15 y + 8 y 2 - 4 y - 10
7. -9m(2m 2 + 3m - 1)
3 2
= 12 y + 2 y - 19 y - 10
= -9m(2m 2 ) - 9m(3m) - 9m(-1)
R.1 Exercises = -18m3 - 27m2 + 9m

1. (2 x 2 - 6 x + 11) + (-3x 2 + 7 x - 2) 8. 6 x(-2 x3 + 5 x + 6) = -12 x 4 + 30 x 2 + 36 x
= 2 x 2 - 6 x + 11 - 3x 2 + 7 x - 2
9. (3t - 2 y )(3t + 5 y )
= (2 - 3) x 2 + (7 - 6) x + (11 - 2) = (3t )(3t ) + (3t )(5 y ) + (-2 y )(3t ) + (-2 y )(5 y )
= -x 2 + x + 9 = 9t 2 + 15ty - 6ty - 10 y 2

2. (-4 y 2 - 3 y + 8) - (2 y 2 - 6 y - 2) = 9t 2 + 9ty - 10 y 2

= (-4 y 2 - 3 y + 8) + (-2 y 2 + 6 y + 2) 10. (9k + q)(2k - q)
= -4 y 2 - 3 y + 8 - 2 y 2 + 6 y + 2 = (9k )(2k ) + (9k )(-q) + (q)(2k ) + (q)(-q)

= (-4 y 2 - 2 y 2 ) + (-3 y + 6 y ) + (8 + 2) = 18k 2 - 9kq + 2kq - q2

= -6 y 2 + 3 y + 10 = 18k 2 - 7kq - q2

3. - 6(2q 2 + 4q - 3) + 4(-q 2 + 7q - 3) 11. (2 - 3x )(2 + 3x )
2
= (-12q - 24q + 18) = (2)(2) + (2)(3x ) + (-3x )(2) + (-3x )(3x )

+ (-4q 2 + 28q - 12) = 4 + 6x - 6x - 9x2

= (-12q 2 - 4q 2 ) = 4 - 9x2
+ (-24q + 28q) + (18 - 12)
12. (6m + 5)(6m - 5)
= -16q 2 + 4q + 6
= (6m)6m) + (6m)(-5) + (5)(6m) + (5)(-5)
4. 2(3r 2 + 4r + 2) - 3(-r 2 + 4r - 5) = 36m2 - 30m + 30m - 25
= (6r 2 + 8r + 4) + (3r 2 - 12r + 15) = 36m2 - 25
= (6r 2 + 3r 2 ) + (8r - 12r ) + (4 + 15)
= 9r 2 - 4r + 19




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,2 Chapter R ALGEBRA REFERENCE

æ2 öæ ö (k + 2)(12k 3 - 3k 2 + k + 1)
13. çç y + 1 z ÷÷çç 3 y + 1 z ÷÷ 18.
èç 5 8 ÷øèç 5 2 ÷ø
= k (12k 3 ) + k (-3k 2 ) + k (k ) + k (1)
æ 2 öæ 3 ö æ 2 öæ 1 ö æ 1 öæ 3 ö
= çç y ÷÷÷çç y ÷÷÷ + çç y ÷÷÷çç z ÷÷÷ + çç z ÷÷÷çç y ÷÷÷ + 2(12k 3 ) + 2(-3k 2 ) + 2(k ) + 2(1)
çè 5 øèç 5 ø èç 5 øèç 2 ø çè 8 øèç 5 ø
æ 1 öæ 1 ö = 12k 4 - 3k 3 + k 2 + k + 24k 3 - 6k 2
+ çç z ÷÷÷çç z ÷÷÷
èç 8 øèç 2 ø + 2k + 2

=
6 2 1
y + yz +
3
yz +
1 2
z = 12k 4 + 21k 3 - 5k 2 + 3k + 2
25 5 40 16
19. ( x + y + z )(3x - 2 y - z )
6 2 æç 8 3 ö÷ 1 2
= y +ç + ÷ yz + z = x(3x ) + x(-2 y ) + x(-z ) + y (3x ) + y(-2 y )
25 çè 40 40 ÷ø 16
6 2 11 1 2 + y(-z ) + z(3x ) + z(-2 y ) + z(-z )
= y + yz + z
25 40 16 = 3x 2 - 2 xy - xz + 3xy - 2 y 2 - yz + 3xz
æ3 öæ ö - 2 yz - z 2
14. çç r - 2 s ÷÷çç 5 r + 1 s ÷÷
çè 4 ÷ç
3 øè 4 3 ø ÷
= 3x 2 + xy + 2 xz - 2 y 2 - 3 yz - z 2
æ 3 ÷öæ 5 ÷ö æ 3 ÷öæ 1 ÷ö æ 2 ÷öæ 5 ÷ö
= çç r ÷÷çç r ÷÷ + çç r ÷÷çç s ÷÷ + çç - s ÷÷çç r ÷÷ 20. ( r + 2s - 3t )(2r - 2s + t )
çè 4 øèç 4 ø çè 4 øèç 3 ø çè 3 øèç 4 ø
= r(2r ) + r(-2s ) + r(t ) + 2s(2r ) + 2s(-2s )
æ 2 öæ 1 ö
+ çç - s ÷÷÷çç s ÷÷÷ + 2s(t ) - 3t (2r ) - 3t (-2s) - 3t (t )
èç 3 øèç 3 ø
15 2 1 5 2 = 2r 2 - 2rs + rt + 4rs + 2st - 6rt + 6st
= r + rs - rs - s 2
16 4 6 9 - 3t 2 + 2rt - st + t 2
15 2 7 2
= r - rs - s 2 = 2r 2 + 2rs - 5rt - 4s 2 + 8st - 3t 2
16 12 9
21. ( x + 1)( x + 2)( x + 3)
15. (3 p - 1)(9 p 2 + 3 p + 1)
= [ x( x + 2) + 1( x + 2)]( x + 3)
= (3 p - 1)(9 p 2 ) + (3 p - 1)(3 p) + (3 p - 1)(1)
= [ x 2 + 2 x + x + 2]( x + 3)
2 2
= 3 p(9 p ) - 1(9 p ) + 3 p(3 p)
= [ x 2 + 3x + 2]( x + 3)
- 1(3 p) + 3 p(1) - 1(1)
= x 2 ( x + 3) + 3x( x + 3) + 2( x + 3)
= 27 p3 - 9 p 2 + 9 p 2 - 3 p + 3 p - 1
= x 3 + 3x 2 + 3 x 2 + 9 x + 2 x + 6
= 27 p3 - 1
= x 3 + 6 x 2 + 11x + 6
2
16. (3 p + 2)(5 p + p - 4)
22. ( x - 1)( x + 2)( x - 3)
= (3 p)(5 p 2 ) + (3 p)( p ) + (3 p)(-4) = [ x( x + 2) + (-1)( x + 2)]( x - 3)
+ (2)(5 p 2 ) + (2)( p) + (2)(-4) = ( x 2 + 2 x - x - 2)( x - 3)
= 15 p 3 + 3 p 2 - 12 p + 10 p 2 + 2 p - 8 = ( x 2 + x - 2)( x - 3)
= 15 p 3 + 13 p 2 - 10 p - 8 = x 2 ( x - 3) + x( x - 3) + (-2)( x - 3)

17. (2m + 1) (4m 2 - 2m + 1) = x 3 - 3x 2 + x 2 - 3x - 2 x + 6

= 2m(4m2 - 2m + 1) + 1(4m 2 - 2m + 1) = x 3 - 2 x 2 - 5x + 6

= 8m3 - 4m 2 + 2m + 4m 2 - 2m + 1 23. ( x + 2)2 = ( x + 2)( x + 2)
= 8m3 + 1 = x( x + 2) + 2( x + 2)
= x2 + 2 x + 2 x + 4
= x2 + 4 x + 4




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,Section R.2 3

24. (2a - 4b)2 = (2a - 4b)(2a - 4b) 4. 60m4 - 120m3n + 50m2n 2
= 2a(2a - 4b) - 4b(2a - 4b) = 10m2 ⋅ 6m2 - 10m2 ⋅ 12mn + 10m2 ⋅ 5n 2
= 4a 2 - 8ab - 8ab + 16b2 = 10m2 (6m2 - 12mn + 5n 2 )
2 2
= 4a - 16ab + 16b
5. m 2 - 5m - 14 = (m - 7)(m + 2)
3
25. ( x - 2 y) since (-7)(2) = -14 and -7 + 2 = -5.
= [( x - 2 y )( x - 2 y )]( x - 2 y )
6. x 2 + 4 x - 5 = ( x + 5)( x - 1)
= ( x 2 - 2 xy - 2 xy + 4 y 2 )( x - 2 y )
since 5(-1) = -5 and -1 + 5 = 4.
= ( x 2 - 4 xy + 4 y 2 )( x - 2 y )
= ( x 2 - 4 xy + 4 y 2 ) x 7. z 2 + 9 z + 20 = ( z + 4)( z + 5)
since 4 ⋅ 5 = 20 and 4 + 5 = 9.
+ ( x 2 - 4 xy + 4 y 2 )(-2 y )
= x 3 - 4 x 2 y + 4 xy 2 - 2 x 2 y + 8 xy 2 - 8 y 3 8. b2 - 8b + 7 = (b - 7)(b - 1)
since (-7)(-1) = 7 and -7 + (-1) = -8.
= x 3 - 6 x 2 y + 12 xy 2 - 8 y 3
9. a 2 - 6ab + 5b2 = (a - b)(a - 5b)
26. (3x + y )3 = (3x + y )(3x + y )2
since (-b)(-5b) = 5b2 and -b + (-5b) = -6b.
= (3x + y )(9 x 2 + 6 xy + y 2 )
= 27 x 3 + 18 x 2 y + 3xy 2 + 9 x 2 y 10. s 2 + 2st - 35t 2 = (s - 5t )(s + 7t )
+ 6 xy 2 + y 3 since (-5t )(7t ) = -35t 2 and 7t + (-5t ) = 2t.
= 27 x 3 + 27 x 2 y + 9 xy 2 + y 3
11. y 2 - 4 yz - 21z 2 = ( y + 3z )( y - 7 z )
since (3z )(-7 z ) = -21z 2 and 3z + (-7 z ) = -4 z.
R.2 Factoring
Your Turn 1 12. 3x 2 + 4 x - 7
Factor 4 z 4 + 4 z 3 + 18z 2 . The possible factors of 3x2 are 3x and x and the
possible factors of –7 are –7 and 1, or 7 and –1. Try
4 z 4 + 4 z 3 + 18z 2 various combinations until one works.
= (2 z 2 ) ⋅ 2 z 2 + (2 z 2 ) ⋅ 2z + (2 z 2 ) ⋅ 9 3x 2 + 4 x - 7 = (3x + 7)( x - 1)
2 2
= (2 z )(2z + 2 z + 9)
13. 3a 2 + 10a + 7
Your Turn 2 The possible factors of 3a2 are 3a and a and the
Factor 6a 2 + 5ab - 4b2 . possible factors of 7 are 7 and 1. Try various
combinations until one works.
6a 2 + 5ab - 4b2 = (2a - b)(3a + 4b)
3a 2 + 10a + 7 = (a + 1)(3a + 7)
R.2 Exercises
14. 15 y 2 + y - 2 = (5 y + 2)(3 y - 1)
1. 7a3 + 14a 2 = 7a 2 ⋅ a + 7a 2 ⋅ 2
= 7a 2 (a + 2) 15. 21m2 + 13mn + 2n 2 = (7m + 2n)(3m + n )

2. 3 y 3 + 24 y 2 + 9 y = 3 y ⋅ y 2 + 3 y ⋅ 8 y + 3 y ⋅ 3 16. 6a 2 - 48a - 120 = 6(a 2 - 8a - 20)
= 3 y( y 2 + 8 y + 3) = 6(a - 10)(a + 2)

3. 13 p 4q 2 - 39 p3q + 26 p 2q 2 17. 3m3 + 12m2 + 9m = 3m(m 2 + 4m + 3)
= 3m(m + 1)(m + 3)
= 13 p 2q ⋅ p 2q - 13 p 2q ⋅ 3 p + 13 p 2q ⋅ 2q
= 13 p 2q( p 2q - 3 p + 2q) 18. 4a 2 + 10a + 6 = 2(2a 2 + 5a + 3)
= 2(2a + 3)(a + 1)




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, 4 Chapter R ALGEBRA REFERENCE


19. 24a 4 + 10a3b - 4a 2b 2 32. 16a 4 - 81b4 = (4a 2 )2 - (9b2 )2
= 2a 2 (12a 2 + 5ab - 2b 2 ) = (4a 2 + 9b2 )(4a 2 - 9b2 )
= 2a 2 (4a - b)(3a + 2b) = (4a 2 + 9b2 )[(2a )2 - (3b)2 ]
= (4a 2 + 9b2 )(2a + 3b)(2a - 3b)
20. 24 x 4 + 36 x3 y - 60 x 2 y 2
= 12 x 2 (2 x 2 + 3xy - 5 y 2 ) R.3 Rational Expressions
= 12 x 2 ( x - y)(2 x + 5 y) Your Turn 1
21. 2
x - 64 = x - 8 2 2 z 2 + 5z + 6
Write in lowest terms .
= ( x + 8)( x - 8) 2z 2 + 7z + 3
z 2 + 5z + 6 ( z + 3)( z + 2)
22. 9m2 - 25 = (3m)2 - (5)2 2
=
2z + 7z + 3 ( z + 3)(2z + 1)
= (3m + 5)(3m - 5)
z+2
=
23. 10 x 2 - 160 = 10( x 2 - 16) 2z + 1

= 10( x 2 - 42 ) Your Turn 2
= 10( x + 4)( x - 4) Perform each of the following operations.
z 2 + 5z + 6 2z 2 - z - 1
24. 9 x 2 + 64 is the sum of two perfect squares. It cannot (a) ⋅
be factored. It is prime. 2 z 2 - 5z - 3 z 2 + 2 z - 3
(z + 2)( z + 3) (2 z + 1)( z - 1)
= ⋅
25. z 2 + 14 zy + 49 y 2 = z 2 + 2 ⋅ 7 zy + 72 y 2 (2 z + 1)( z - 3) ( z + 3)( z - 1)
= ( z + 7 y )2 ( z + 2) ( z + 3) (2 z + 1) ( z - 1)
=
(2z + 1) ( z - 3) ( z + 3) ( z - 1)
26. s 2 - 10 st + 25t 2 = s 2 - 2 ⋅ 5st + (5t )2
z+2
= ( s - 5t )2 =
z-3
a-3 5a
27. 9 p 2 - 24 p + 16 = (3 p )2 - 2 ⋅ 3 p ⋅ 4 + 42 (b) +
2 2
a + 3a + 2 a -4
= (3 p - 4)2
a-3 5a
= +
28. a 3 - 216 = a 3 - 63 (a + 2)(a + 1) (a - 2)(a + 2)
a-3 (a - 2)
= (a - 6)[(a )2 + (a )(6) + (6)2 ] = ⋅
(a + 2)(a + 1) (a - 2)
= (a - 6)(a 2 + 6a + 36) 5a (a + 1)
+ ⋅
(a - 2)(a + 2) (a + 1)
29. 27r 3 - 64s 3 = (3r )3 - (4s )3
( a 2 - 5a + 6) + (5a 2 + 5a )
= (3r - 4s )(9r 2 + 12rs + 16s 2 ) =
( a - 2)( a + 2)( a + 1)
30. 3m3 + 375 = 3(m3 + 125) 6a 2 + 6
=
= 3(m3 + 53 ) ( a - 2)( a + 2)( a + 1)

= 3(m + 5)(m2 - 5m + 25) 6(a 2 + 1)
=
( a - 2)( a + 2)( a + 1)
31. x 4 - y 4 = ( x 2 ) 2 - ( y 2 )2
R.3 Exercises
= ( x 2 + y 2 )( x 2 - y 2 )
5v 2 5⋅v⋅v v
= ( x 2 + y 2 )( x + y )( x - y ) 1. = =
35v 5⋅7⋅v 7

25 p 3 5⋅5⋅ p⋅ p⋅ p 5p
2. 2
= =
10 p 2⋅5⋅ p⋅ p 2


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