factoring handout a general factoring strategy · 2013-12-15 · factoring handout a general...
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This Factoring Packet was made possible by a GRCC Faculty Excellence grant by Neesha Patel and Adrienne
Palmer.
FACTORING HANDOUT
A General Factoring Strategy
It is important to be able to recognize the different types of polynomials and know each ones factoring method.
Use these steps to guide you:
1) Factor out the greatest common factor (GCF),if there is one.
2) Are there two terms? (Binomial)
Is it Difference of two squares? If yes, factor by using:
( )( ) Page 2
Note: You cannot factor a binomial in the form .
3) Are there three terms? (Trinomial)
Is it a perfect square trinomial? If yes use:
( ) Page 3
( )
4) Is the form where ? Factor by Product-Sum Method . (Page 5)
5) Is the form where ? Factor by Guess and Check, the ac-Grouping
or one of the other methods attached. (Page 6 and 7)
6) If you can’t factor it by any method above, the polynomial is irreducible. It is prime.
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Factoring Binomials
Difference of Two Squares: ( )( )
Example: Factor
*Notice that both and 9 are perfect squares: and
So ( )( )
Example: Factor
Factor out the GCF first: ( )
*Notice that both and are perfect squares: ( ) and
( )
So ( ) ( )( )
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Factoring Trinomials
Perfect Square Trinomials: ( )
( ) _______________________________________________________________________
Example: Factor
*Notice that both and 9 are perfect squares. So a good first guess at how to
factor this trinomial would be to use their roots:
( )( )
Then we can just work on figuring out what signs need to go in each parentheses.
With a little trial and error, we see that a minus sign in each parentheses would
work.
( )( )
So ( )( )
( )
Another way that we could have looked at this factoring problem would be to
notice that and and ( )( ) [if we are trying to match
things up with the special factoring patterns for perfect square trinomials, then
( )( ) ].
Recognizing the special factoring pattern ( ) , we could
have factored immediately into the form ( )
Example: Factor
*Notice that both and 25 are perfect squares. So a good first guess at how to
factor this trinomial would be to use their roots:
( )( )
Then we can just work on figuring out what signs need to go in each parentheses.
With a little trial and error, we see that a plus sign in each parentheses would
work.
( )( )
So ( )( )
( )
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Another way that we could have looked at this factoring problem would be to
notice that and and ( )( ) [if we are trying to
match things up with the special factoring patterns for perfect square trinomials,
then ( )( ) ].
Recognizing the special factoring pattern ( ) , we could
have factored immediately into the form ( )
Example: Factor
*With a little practice, you may notice that ( )( )
So using the special factoring pattern ( ) we get
( )
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FACTORING TRINOMIALS OF THE FORM cbxax 2 ( )
USING THE PRODUCT-SUM METHOD
(Use when a=1)
Example: 652 xx
STEPS 1. Setup the binomial factors and enter the first term of each factor. Remember, you’re
doing the reverse of FOILING.
(x )(x )
2. Write the value of “b” and “c”: b = 5 , c = 6
3. List all pairs of integers whose product is c.
C= 6
32
61
4. Choose the pair whose sum is b:
b = 5
(This one)
5. Plug the matching pair into the binomial factors:
)3)(2( xx
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FACTORING TRINOMIALS IN THE FORM
FACTOR BY GUESS AND CHECK
Use this method if the a and c values are small or prime.
E.g. Factor
Think reverse FOILing. The only choice for the first terms in each binomial is 5x and x to obtain
the product of that appears in the first term above.
( )( )
We wish to obtain the c value of 2 when we FOIL back. Our factors of 2 are 1 and 2. So either we
have:
( )( ) Or
( )( )
FOILing the first option gives the middle term of that appears in the original trinomial.
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FACTORING TRINOMIALS IN THE FORM
FACTOR BY THE “ac” AND GROUPING METHOD
Use when or 0:
Example 1:
STEPS: 1. Factor out a GCF if there is one. This example does not have one.
2. Then use the steps below to factor the trinomial into two binomial factors.
3. List the values of a, b and c in the expression:
4. Find the product of “ ”:
5. List the factor pairs that give the product of :
6. Find the pair of factors whose sum equals “b”, and write as (i.e. The middle term including the variable)
7. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms:
8. Use the Grouping Method to complete the factoring as follows: Group the first two terms together and the last two terms together:
( ) ( ) 9. Factor out any common factors from the first group and any from the second group:
( ) ( )
Ist term 2nd term
Notice that we now have an expression with just 2 terms. Each term should have a common factor (2x + 1 in this case).
10. Factor out this common factor from each term: ( )( ). These are your binomial factors.
11. FOIL out to double check that your factors match the original equation.
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FACTOR BY THE “ac” AND GROUPING METHOD continued:
Example 2:
1. Factor out the GCF: ( )
2. Ignore the GCF for now. We wish to factor . List the values of a, b and c for this quadratic expression:
a = 2, b = -15, c = -27
3. Find the product of “ ca ”: ( )
4. List all factor pairs of the product in step 3. Be systematic and keep going until you find
the pair whose sum equals (the value of b). Notice that the signs of the factors must
be opposites:
( )( ) ( )( ) for example. The two factors we want are 3 and
Rewrite the middle term using these 2 factors:
5. Replace these two terms for bx in the original expression, so that you now have an expression with 4 terms:
6. Group the first two terms together and the last two terms together: ( ) ( )
7. Factor the GCF from each set of parentheses: ( ) ( ) Notice that we now have an expression with just two terms.
Each term has a common factor of . 1st term 2nd term
8. Factor out this common factor from each term to obtain your two binomial factors: ( )( )
Note: If you had reversed the two middle terms in step 5 to obtain Be careful how you handle the parentheses. If you have a minus outside the second set of parentheses, you will need to change the sign of every term inside the parentheses as follows:
( ) ( ) Both the 18x and 27 change signs.
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FACTORING TRINOMIALS IN THE FORM cbxax 2
USING THE TABLE METHOD
Use when 1a or 0:
STEPS:
1. Example: 372 2 xx
2. Factor out a GCF if there is one. Then use the steps below to factor the remaining trinomial.
3. List the values of a, b and c in the quadratic expression: a = 2, b = 7, c = 3
Setup a box as shown below. Write the value of in the top left unshaded box and c bottom right
unshaded box.
3. Above example
5. Find the product of “ac”:
632
6. List factors of the product in step 5: 61 , 23
7. Find the pair of factors whose sum equals “b”, and write as
xx 61 (i.e. The middle term, i.e. include the variable)
8. Plug these two terms into the two unshaded empty boxes in the table. (it doesn’t matter
which term goes into which box). Then factor out the common factors in each row and column and place
these in the shaded boxes:
Factor
First term 2ax
Last term c
Factor
22x
3
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9. The two shaded boxes give you the factored binomials products:
)3)(12( xx
FOIL out to check you get the original trinomial expression.
Factor x 3
2x 22x 6x
1 x 3
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Practice Problems
Begin by doing all of the problems that end with a 7 (problems 7, 17, 27, etc.). Check your answers by
multiplication; if you multiply your answer out and simplify, you should get the original polynomial. For problems
that you have trouble with, work on the other nine problems in that group. Again, check your answers by
multiplication.
Group A
1. 1072 xx
2. 892 xx
3. 1272 nn
4. 30112 aa
5. 24102 zz
6. 1282 tt
7. 1892 xx
8. 36152 xx
9. 18112 xx
10. 30132 mm
Group B
11. 122 xx
12. 3242 yy
13. 1522 zz
14. 22 1211 yxyx
15. 822 nn
16. 3652 mm
17. 2452 xx
18. 422 aa
19. 2422 xx
20. 202 tt
Group C
21. 253 2 xx
22. 4125 2 yy
23. 492 2 aa
24. 3103 2 nn
25. 594 2 zz
26. 22 374 yxyx
27. 5112 2 xx
28. 299 2 tt
29. 384 2 ww
30. 10197 2 xx
Group D
31. 32 2 xx
32. 23 2 tt
33. 584 2 tt
34. 22 34 nmnm
35. 869 2 qq
36. 656 2 ww
37. 443 2 nn
38. 22 274 yxyx
39. 103 2 yy
40. 235 2 tt
12
Group E
41. 203112 2 xx
42. 22 102718 zyzy
43. 155620 2 aa
44. 356224 2 tt
45. 188936 2 nn
46. 21256 2 xx
47. 20296 2 yy
48. 22 152812 xyxy
49. 183920 2 mm
50. 3010930 2 xx
Group F
51. 252012 2 xx
52. 211912 2 yy
53. 22 1220 nmnm
54. 181124 2 tt
55. 3020 2 tt
56. 36910 2 xx
57. 22 242312 yxyx
58. 303112 2 cc
59. 22 302340 baba
60. 101912 2 xx
Group G
61. 92 b
62. 814 2 z
63. 22 12136 ts
64. 25144 2 x
65. 3250 2 x
66. 649 2 a
67. 22 4916 yx
68. 10081 2 n
69. 22564 2 y
70. 814 x
Group H
71. 25309 2 xx
72. 49284 2 nn
73. 22 498436 zyzy
74. 64489 2 aa
75. 11025 2 xx
76. 43681 2 cc
77. 16249 2 xx
78. 25204 2 yy
79. 817216 2 xx
80. 6411249 2 bb
Group I
81. 82812 2 aa
82. zzz 15164 23
83. 123012 2 nn
84. ttt 245412 23
85. 234 102515 xxx
86. 22 1248 nmnm
87. 202530 2 zz
88. 2010100 2 xx
89. yyy 30912 23
90. 186048 2 xx
13
Group J
91. 121812 2 xx
92. 10019590 2 yy
93. 223 248248 abbaa
94. 22 123624 yxyx
95. ttt 123120 23
96. www 243624 23
97. xxx 458736 23
98. 2406448 2 xx
99. 3223 24212 xyyxyx
100. 1805472 2 mm