Sect 1.4 pg 44

CHAPTER
',
1
ALGEBRAIC EXPRESSIONS
VV V V V V V VV V V V V V V VVV V V V V V V V
1. When a student was directed to factor the polynomial 4x3ys - 8*'yo completely on a test, she
responded with2x2y3(2xy2 - 4y). When her teacher did not give her full credit she complained,
polynomial is obtained. Was her
.4
2.
EXefeiSeS V V V V V V V V V V
indicating that when her answer is multiplied out, the original
teacher justifled in doing this? Why or why not?
Why does the binomial x6 64 fall into /wo special categories discussed in this section?
-
Factor the greatest common factor from the polynomial' See Example 1'
3. 4k2m3 *
8kam3
-
4. 28ras2 I 7r3s -
l2k2ma
5. 2(a + b) + 4m(a + b)
7. (5r - 6)(r + 3) - (2r - 1)(r +
9.2(m - 1) - 3(m * r)'+ 2(m -
35ras3
e.46-2)2+3(y-2)
8.(32+2)k+4-k+6)k+a)
10. 5(a + 3)' - 2(a + 3) + (a + 3)2
3)
t)3
Factor the polynomial by grouping. See Example 2'
+ 9t -
- 15
13.2mal6-am4-3a
15. 2Oz2 a 18x2 - 8zx -
l1^. 6st
12. l\ab - 6b + 35a - 2l
14. 15 - 5m2 - 3r2 * m2r2
10s
45zx
16. Shalita factored l6a2 - 40a - 6a + 15 by grouping and obtained (8a - 3)(2a - 5)' Jamal
factored the same polynomial and gave an answer of (3 - 8a)(5 - 2a). Are both of these
answers correct? If not, whY not?
Factor the trinomial. See Example 3.
17. 6a2
-
48a
-
20.9ya-54y'+45y2
- lab - 6b2
26. 3Oa2 * am - m2
23. 5a2
19.3m3+12m2*9m
-l Llmr - l5r2
- 320
21. 6k2 + 5kp - 6p2
24. l2s2 -l llst - 5t2
22. l4m2
27.24aa+loa3b-Aa',b2'
28. 18xs
I
15xaz.
31.32a2
-
48ab
18. 8h2
120
-
24h
25.9x2-6x3+xa
-
J5x3z2
Factor the perfect square trinomial. See Example 4'
* lzm -l 4
32. 2Op2 - l\Opq + 125q2
35. (a - 3b)2 - 6(a - 3b) + 9
29. 9m2
+ 25
+ 49
+
28xy
4x2y2
33.
30. l6p2
-
4Op
36. (2p
34.9m2n2
+ q)' -
t0(2P
+
ct)
+
-
+ l8b2
lZmn
*
4
-
81)o
25
Factor the dffirence of two squares. See Example 5'
3g. l6q2 - 25
42. (p - 2q)2 *
- 16
41. (a + b)2 - 16
37. ga2
39. 25s4
100
-
9t2
- 625
45. Which of the following is the correct complete factorization of -ra - 1?
(a) (x'- l)(x'z + 1) (b) (r'+ 1)(x + 1)(x - 1)
(d) (x - 1)2(x + l)'z
(c) (x2 - 1)2
43' p4
46. Which of the fotlowing is the correct factorization of 'r3
+
4)
4)
4)
(a) (x + 2)3
(c) (x + 2)(x2
-
2x +
(b) (x + 2)(x2 + 2x +
(d) (x + 2)(*' - 4x +
8?
40. 3622
44. m4
- 8l
1,4 FACTORING
7YY
45
Factor the sum or difJerence of cubes. See Example 6.
17.8 - a3
5l.27ye + 72526
55. 21 - (4 -t 2n)3
+ zl
52. 2'723 + 72gy3
48. 13
56. 125
-
(4a
-
49. 125x3 - 27
53. (r + 6)3 - 216
50. 8m3
s4. (b
+
-
27n3
3)3
-
2l
b)3
polynomial x6 - I can be considered either a dffirence of squares or a difference
of cubes.
Work Exercises 57-62 in order, and see some interesting connections between the
results
obtained when two dffirent methods of factoring are used.
The
57. Factor-x6
- 1 by flrst factoring as the difference oftwo squares, and then factor further by using
the patterns for the sum of two cubes and the differencJof two cubes.
58. Factor x6
- 1 by f,rst factoring as the difference of two cubes, and then factor further by using
the pattern for the difference of two squares.
59. Compare your answers in Exercises 57
of.ra+x2+1?
and 58. Based on these results, what is the factorization
+ x2 * 1 cannot be factored using the methods described in this section.
However, there is a technique that allows us to factor it. This technique is shown below.
Supply
the reason that each step is valid.
60. The polynomial x4
xa+x2*l:xat2x2+l_
x2
:(*af2x2+11 _rz
: (_r2 f l)2 _ *z
: (x2 f I _ x)(x2 + 1 +
:(.x2-x+l)(x2+x+1)
x)
61. Compare your answer in Exercise 59 with the final line in Exergise 60. What do you notice?
62. Factor x8 + xa * 1 using the technique outlined in Exercise 60.
Factor each polynomial, using the method of substitution. See Example
63. ma
-
66. 6(42
3m2
- :)'
-
lo
- 2a2 - 48
67. 9(a - 4), + 30(a -
7.
64, aa
+ i(42
-
3)
- :
4) +
25
6s. 7(3k - 1), + 26(3k - 1) 68.20(4* p)'-3@- p)-2
8
Factor by any method.
69. 4b2
+ 4bc -t
72. 8r2 - 3rs
75. 422 + 2gz
c2
-
*
10s2
+
4g
76
78.b2+gb-t16-a2
81. 72m2 -l 76mn - 35n2
84.
10012
87. (x
-
l6gs2
+ y)' -
(-r
-
y)'
- r), * aQy - 1) + 4
73. patm 2nt -t qlm - 2n)
76. 6pa + 7p2 - 3
79. 125m6 - 216
70. (2y
+ 125q3
85. 14422 + t21
82.216p3
88. 4za
-
jz2
-
15
71.x2+xy-5x-5y
74, 36a2 + 6Oa + 25
77. 1000x3 + 343y3
80.q2+6q+9-p2
83.4p2+3p-l
86. (3a + 5)' - l8(3a + 5) + 81
46
CHAPTER
1
ALGEBRAIC EXPRESSIONS
Factor each polynomial. Assume that all yariables used in exponents represent positive integers.
See Example 5(d).
+
- 6*
94. 25x4' - 2Ox2' +
- z'xb - x2b
93. 4y2" - 12y" + 9
g0,6z2n
- 6s2q
25x4,
92, 16y2, -
gg. 12
rsq
91.9otr<
4
95. Explain how the accompanying {igures give geometric interpretation to the formula
*' - y': (* + l)(x -
Y).
)
.J-*
tffi'
I
t-'
x*)
x
96. Explain how the accompanying figures give geometric interpretation to the formula
x2+zxyl-Y':(r+Y)'.
l,{r---l
r[[T..-l],
,lll,lT-----_llll
'll
,L--------r
I
II
_L
,J[l
u i
I I
iI lI
lf.
lt'
I]
T_i_
Find a value of b or c that will make the polynomial a perfect square trinomial'
97.
422
1.5
+ bz + 81
98. 9p2
Rational ExPressions
+
bp
+ 25
99. lO0r2 - 60r -t
c
100. 49xz
*'70x
*
c
vvv
An expression that is the quotient of two algebraic expressions (with denomina-
a fractional expression. The most common fractional expresof two polynomialsl these are called rational expresquotients
sions are the
expressions involve quotients, it is important to keep
fiactional
sions. Since
that satisfy the requirement that no denominator
variable
of
the
track of values
in
x
*
be 0. For example,
-2 the rational expression
tor noi 0; is called
x*6
x-l
2

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