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1 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.

FACTORING HANDOUT A General Factoring Strategy · 2013-12-15 · FACTORING HANDOUT A General Factoring Strategy ... 2x2 x 3 32. 3t2 t 2 33. 4t2 8t 5 34. 4m 2 3mn n 2 35. 9q 2 6q 8

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Page 1: FACTORING HANDOUT A General Factoring Strategy · 2013-12-15 · FACTORING HANDOUT A General Factoring Strategy ... 2x2 x 3 32. 3t2 t 2 33. 4t2 8t 5 34. 4m 2 3mn n 2 35. 9q 2 6q 8

1

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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2

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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3

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

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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

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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