2.03
Factoring the Common Factor
from Basic Algebra: One Step at
a Time © 2002
P.
139-146
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
In the last
section, you learned to factor numbers into prime factors. In that
section, you were asked to factor a number. If, for example, you
were asked to factor 15, you would quite naturally write "3 times 5" or
"5 times 3"! The key word was "times," which means multiplication.
When asked to factor a given number, you naturally answer with a product
of two numbers. The following is a working definition of
Afactor@.
WORKING DEFINITION:
TO FACTOR: to EXPRESS AS A
PRODUCT (MULTIPLICATION!)
|
While there are
actually many different types of factoring, especially in higher
mathematics courses, most of this chapter will focus on two main types
of factoring. For this reason, we will call this the
AFactoring
Two Step.@
Because so many future topics in algebra require you to be able to
factor expressions, it is probably safe to say that these next sections
and indeed the rest of this chapter will be some of the most important
lessons in the entire book! Please spend an appropriate amount of
time, even more than usual, on these sections!
The first step in
any factoring problem is to
Afactor
the common factor.@
Factoring the common factor is simply using the distributive property in
reverse. Study the examples below.
EXAMPLES of DISTRIBUTIVE PROPERTY
EXAMPLES of FACTORING:
6(x + 7) = 6x + 42
6x + 42 = 6(x
+ 7)
7(2x + 3) = 14x + 21
14x + 21 = 7(2x
+ 3)
9(x − 4) = 9x
− 36
9x − 36
= 9(x − 4)
12(2x + 1) = 24x + 12
24x + 12 = 12(2x
+ 1)
5(3x − 2y + 4) =
15x − 10y + 20
15x
− 10y + 20
= 5(3x − 2y + 4)
5x(x + 4) = 5x2
+ 20x
5x2
+ 20x = 5x(x + 4)
When factoring the
common factor, look for a number or variable that divides into both (or
all) terms. If there is more than one common factor, be sure to get the
largest common factor you can find. First write down the common factor.
Then, open parentheses, and put down all the other factors that are left
over.
EXAMPLE
1. 12x + 36
SOLUTION: There are
several numbers that divide evenly into both 12 and 36: 1,2,3,4,6, and
12. Take the largest common factor, which is 12. Write down the 12, then
open parentheses: 12 ( + ). In the parentheses you put the
remaining factors, x and 3, like this:
12x + 36 = 12 ( + )
= 12 ( x + 3 )
EXERCISES:
Factor completely.
1. 3x
+ 15
2. 8x + 24
3. 7x − 28
= 3(____ +
____) = 8(____ + ____)
= _____(____ −
____)
4. 5x
− 25
5. 34x + 17y
6. 14x + 28y
= _____(
− )
= 17 ( )
= _____( )
7. 12x +
36y 8. 15x + 60y
9. 7x + 7
=
_____( ) = ____(
)
= _____( + 1)
10. 3x2
+ 3 11. 42x2
+ 21
12. 30x2
+ 15
= 3 ( ____ +
____ ) = 21 ( )
= _____( )
13. 5x2
+ 15x 14. 7x2
+ 21x
15. 7x2
+
14x
= 5x (
x + ) = 7x
( )
= _____( )
16. 7x2
− 14x
17. 3x2
+ 12x
18. 21x2
+ 30x
=
_____( ) =
_______________
= _________________
19. 16x2
− 18x 20. 12x2
− 30x
21. x3
+ 3x2
=
_____( )
= _____( )
= x2(
)
22. x3
+ 4x2
23. x3
+ 4x
24. x3
− 4x2
=
_________________ = ________________
= __________________
25. 4x3
+ 8x2
26. 4x3
+ 8x
27. 12x3
− 8x2
= 4x2(
) = 4x ( )
= __________________
28. 16x3
− 24x2
29. 16x3
+ 32x2
30. 12x3
+ 18x2
= 8x2(
)
= _____( )
= __________________
31. 12x3
− 18x2
32. 45x3
+ 30x2
33. 12x2
+ 12
=
_________________ = _________________
= __________________
34. 24x2
+ 12 35. 24x2
+ 12x
36. 144x2
+ 12x
=
_________________ = _________________
= __________________
37. 16x2
+ 48x3
38. 16x2
− 48x3
39. 24x4
+ 36x3
= 16x2
( ) =
__________________
= __________________
40. 24x3
+ 24x2
41. 24x3
+ 24x
42. 24x4
+ 16x2
=
________________ = __________________
= __________________
43. 6x +
9y − 12
44. 3x + 6y + 12z
45. 9x + 18y + 9
= 3(____ + ____
− ____)
= ____( )
= 9( )
46. 30x +
20y + 10
47. 30x + 20y
− 5
48. 35x + 28y − 14z
= 10(____ + ____
+ ____)
= ____________________
= ___________________
49. 30x2
+ 20xy − 10y2
50. 30x3
+ 20xy + 10x2
=
________________________
= _______________________
51. 12x3
+ 24x2
+ 24x 52. 12x3
− 24x2
+ 3x
=
________________________ =
_______________________
53. 19x3
+ 19x2y
+ 38x2
54. 36x3
+ 24x2y
+ 12x2
=
_________________________ =
_______________________
55. 16x2
+ 32x3
56. 16x3
+ 32x2
= 16x2(
)
= ____________________
57. 16x2
−
12x3
58. 16x3
−
12x2
=
____________________
= ____________________
59. y5
−
14y3
60. x10
+ 5x3
= y3(
)
= x3( )
From exercises
#55 − 60 above, observe the rule listed below!
RULE
When factoring powers, take out the lowest
exponent (power) of the factor. Then subtract exponents.
|
61. 16x2y3
−
12x3y2
62. 4x3y3
+ 8x2y4
= 4x2y2(
) = 4x2y3( )
63. 5x3y3
+ 10x2y
64. 8x5y2
− 16x4y3
=
__________________ =
____________________
65. 8x5y3
+ 12x3y4
66. 8x3y4
+ 24x2y6
=
__________________ =
____________________
When an expression
contains negative terms, it is sometimes helpful to be able to
Afactor
out the negative@
or to factor out a
A−1".
If you factor out a
A−1",
this changes each sign inside the parentheses that follow, as in the
following example.
EXAMPLE 2.
Factor out a
A−1"
from the expression −4x − 6y + 9.
Solution: −4x
− 6y + 9 = −1(4x + 6y − 9) or
=
− (4x + 6y − 9)
This can be verified
by using the distributive property!
EXERCISES.
In each of the following, factor completely, including the
Anegative.@
67.
− x2
− 4x + 6 68. − x2
+ 5x
− 1 69. − x2
+ 3x + 7
=
−1( + −
) = −1( ) =
____________________
or
− ( + −
) or − ( )
70.
− 6 x2
+ 12 71. − 6x − 9y + 15
72. − 8x + 20y − 24
= − 6
( ) = − 3 ( )
= _____________________
73.
− x2
− 4x
74. − x2
− 7x 75. − x2
+ 7x
= − x
( ) = __________________
= _____________________
76.
− 4x + 12x2
77. − 8x − 12x2
78. − 4x2
− 4y2
= − 4x ( 1
− _______ ) = ___________________ =
_____________________
79.
− x2
− 3x + 8xy 80. − 4x2
− 16x + 16 81. − 8 + 8x − x2
=
__________________ = ___________________ =
_____________________
82.
− 8 − 8x − 8x2
83. 8 − 16x − 40x2
84. 35 + 20x − 5x2
=
________________ = ________________ =
________________
In each of the
following exercises, factor the common factors. As you do, observe
how you move from the simple to the more complicated; from the concrete to
the abstract.
85a) yx
+ 7x 86a) 4xy
+ 3y
= x
( ) =
______________
b) ya +
7a
b) 4xa + 3a
= a
( ) =
______________
c) y$ +
7$ c) 4x$ +
3$
= $
( ) =
______________
d) y (Junk)
+ 7 (Junk) d) 4x (Junk) + 3
(Junk)
= (Junk)
( ) = ______________
e) y (x+4)
+ 7 (x+4) e) 4x
(y−7) + 3 (y−7)
= (x+4)
( ) = ______________
87. a (x+4)
− 5 (x+4)
88. 5a (b+7) + 3 (b+7)
89. 5a (3x+4)
+ 9b(3x+4) 90. 10a (9y−7)
− 3 (9y−7)
RULE
In order to factor a common factor, you
must have an identical factor common to all terms.
Be sure to count terms first.
|
EXAMPLE
3. Can you factor 5u(3x+4) + 9v(3x−4)
in this manner?
[See answer below!]
EXAMPLE
3 Answer!NO!
There is no factor common to both terms.]
EXERCISES. Factor the common factor in each of the following expression.
91. x(x−y)
− y(x−y) 92. x(x−y)
− y(x−y) + 4(x−y)
93. x(x−y)
+ y(x−y) − 4(x−y) 94. x(2x+3y)
− y(2x+3y) + 4(2x+3y)
EXTRA CHALLENGE:
95. (x + y)2
− z(x +
y) 96. (x − y)2
− z(x − y)
= ( )[( )
− _____ ]
= ( ) ( )
97. (x − y)2
−
y(x − y) 98. (x +
y)2
− y(x + y)
99. (2x + 3y)2
− 5(2x + 3y)
100. (2x − 3y)2
− 5(2x − 3y)
101. (2x + 3y)2
+ 5(2x + 3y) 102. (x + y)2 + (x
− y)(x + y)
ANSWERS 2.03
p. 140-145:
1.
3(x+5); 2. 8(x+3); 3. 7(x-4); 4.
5(x-5); 5. 17(2x+y); 6. 14(x+2y);
7. 12(x+3y);
8. 15(x+4y); 9. 7(x+1);
10. 3(x2+1);
11. 21(2x2+1);
12. 15(2x2+1);
13. 5x(x+3); 14. 7x(x+3); 15. 7x(x+2);
16. 7x(x-2); 17. 3x(x+4);
18. 3x(7x+10); 19. 2x(8x-9); 20. 6x(2x-5);
21. x2(x+3);
22. x2(x+4);
23. x(x2+4);
24. x2(x-4);
25. 4x2(x+2);
26. 4x(x2+2); 27.
4x2(3x-2);
28. 8x2(2x-3);
29. 16x2(x+2);
30. 6x2(2x+3);
31. 6x2(2x-3);
32. 15x2(3x+2);
33. 12(x2+1);
34. 12(2x2+1);
35. 12x(2x+1); 36. 12x(12x+1); 37. 16x2(1+3x);
38. 16x2(1-3x);
39. 12x3(2x+3);
40. 24x2(x+1);
41. 24x(x2+1);
42. 8x2(3x2+2);
43. 3(2x+3y-4); 44. 3(x+2y+4z);
45. 9(x+2y+1); 46. 10(3x+2y+1); 47. 5(6x+4y-1);
48. 7(5x+4y-2z); 49. 10(3x2+2xy-y2);
50. 10x(3x2+2y+x);
51. 12x(x2+2x+2);
52. 3x(4x2-
8x+1); 53. 19x2(x+y+2);
54. 12x2(3x+2y+1);
55. 16x2(1+2x);
56. 16x2(x+2);
57. 4x2(4-3x);
58. 4x2(4x-3);
59. y3(y2-14);
60. x3(x7+5);
61. 4x2y2(4y-3x);
62. 4x2y3(x+2y);
63. 5x2y(xy2+2);
64. 8x4y2(x-2y);
65. 4x3y3(2x2+3y);
66. 8x2y4(x+3y2);
67. -(x2+4x-6);
68. -(x2-5x+1);
69. -(x2-3x-7);
70. -6(x2-2); 71.
-3(2x+3y-5); 72. -4(2x-5y+6); 73. -x(x+4);
74. -x(x+7); 75. -x(x-7); 76. -4x(1-3x);
77. -4x(2+3x); 78. -4(x2+y2);
79. -x(x+3-8y); 80. -4(x2+4x-4);
81. -(8-8x+x2);
82. -8(1+x+x2);
83. -8(-1+2x+5x2);
84. -5(-7-4x+x2);
85a) x(y+7); b) a(y+7);c) $(y+7);d) (Junk)(y+7);e)
(x+4)(y+7);
86a) y(4x+3); b)
a(4x+3); c) $(4x+3);d) (Junk)(4x+3);e) (y-7)(4x+3);
87. (x+4)(a-5); 88.
(b+7)(5a+3); 89. (3x+4)(5a+9b); 90. (9y-7)(10a-3); 91. (x-y)2;
92. (x-y)(x-y+4);
93. (x-y)(x+y-4); 94. (2x+3y)(x-y+4);
95. (x+y)(x+y-z); 96. (x-y)(x-y-z);
97. (x-y)(x-2y);
98. x(x+y); 99. (2x+3y)(2x+3y-5);
100. (2x-3y)(2x-3y-5); 101. (2x+3y)(2x+3y+5);
102. 2x(x+y).
Return to main page
Math in Living
C
O
L O
R
!!
Return to Basic
Algebra page
Return to
Intermediate Algebra page
Return
to College Algebra page