REVIEW EXERCISES.
Use the F OI L method to multiply each of the following:
1.
(x − 2)(x + 2)
2. (x − 3)(x + 3)
3. (x − 1)(x + 1)
4. (x
− 6)(x + 6) 5. (2x
− 3)(2x + 3)
6. (3x − 5)(3x + 5)
7. (x
+ 2)2 8. (x
+ 5)2 9. (x
− 5)2
=(x
+ 2)(x + 2)
= =
=____+____+____ =
=
10. (x
− 7)2 11. (2x
+ 3)2 12. (3x
− 5)2
As in the previous
section, in each of these exercises, you were given a
product of two binomials. Now, as before, the problem will be to
reverse the process, and factor the difference of two squares or
factor a perfect square trinomial.
EXERCISES. In each of
the following, factor the difference of squares:
1.
x2 − 9
2. x2 − 16
3. x2 − 49
(x −
___)(x + ___) (x −
___)(x + ___) (x
− ___)(x + ___)
4. x2
− 64 5. x2
− 100 6. x2
− 81
(x −
___)(x + ___)
7. x2
− 25 8. x2
− 121 9. x2
− 144
10. x2
− 169 11. x2 −
a2 12. x2
− b2
13. 4x2
− 9 14. 9x2
− 16 15. 16x2
− 49
(2x −
___)(2x + ___) (3x − ___)(3x
+ ___) (___ − ___)(___+___)
16. 25x2
− 64 17. 9x2 −
100 18. 49x2
− 81
19. 9x2
− 25y2 20. 49x2
− 121y2 21. 25x2
− 144a2
22. 16x2
− 81b2 23. 9y2 − 169x2
24. 36y2 − 49x2
Don't forget that the
first step in any factoring problem is to factor the common factor first.
This means that each of the following exercises requires two steps. Again,
this is the "factoring two-step."
25. 9x2
− 9 26. 3x2
− 12 27. 5x2
− 45
9(
) 3( )
___( )
9(
)( ) 3( )(
) ___( )( )
28. 4x2
− 64 29. 4x2
− 100 30. 8x2
− 72
31. 25x2
− 25 32. 7x2
− 700 33. x3
− 16x
34. 3x3
− 75x 35. 5x3
− 80x 36. 2x3
− 50x
37. 16x4
− 16x2y2 38.
9y4 − 9x2y2
39. 81y4 − 9x2y2
40. 64x4
− 4x2y2 41.
12y4 − 12x2y2
42. 79y4 − 79x2y2
Factor each of the
following perfect square trinomials:
43. x2
+ 4x + 4 44. x2
+ 6x + 9 45. x2
+ 2x + 1
=(
)( )
=
( )2
46. x2
+ 12x + 36 47. x2
+ 14x + 49 48. x2
+ 20x + 100
49. x2
− 8x + 16 50. x2
− 4x + 4 51. x2
− 10x + 25
52. x2
− 12x + 36 53. x2
− 18x + 81 54. x2
− 24x + 144
Remember to factor the
common factor first:
55. 5x2
− 20x + 20 56. 2x2
− 20x + 50 57. 3x2
+ 6x + 3
____( )
____(
)( )
____( )2
58. 3x2
− 60x + 300 59. 6x2
− 36x + 54 60. 4x2
+ 32x + 64
61. x3
+ 4x2 + 4x
62. x3 − 12x2 + 36x
63. 9x3 − 18x2
+ 9x
64. 6x4
− 36x3 + 54x2
65. 12y4 − 48y3 + 48y2
66. 2y4 − 28y3 + 98y2
In the next exercises,
remember that sum of squares, such as x2+9 or
x2+4, does NOT factor by this method.
67.
x4 − 16
68. x4
− 1
= (x2
− ___)(x2 + ___)
= (x
− ___)(x + ___)(x2 + ___)
69. x4
− 81 70. x4
− y4
71. 81x4
− 16y4 72. 16x4
− 81
73. x4
+ 10x2 + 9
74. x4 + 13x2 +
36
75. x4
− 13x2 + 36 76.
x4 − 5x2 + 4
77. x4
− 29x2 + 100 78.
y4 + 5y2 − 36
79. y4
− 8y2 + 16 80. y4
− 18y2 + 81
ANSWERS 2.05
p.157:
1.
x2 - 4; 2. x2 -
9; 3. x2 - 1; 4. x2
- 36; 5. 4x2 - 9; 6. 9x2
- 25; 7. x2+4x+4;
8. x2+10x+25;
9. x2-10x+25; 10.
x2-14x+49; 11. 4x2+12x+9;
12. 9x2-30x+25.
p. 158-163:
(NOTE: Factors may be given in any order!)
1. (x-3)(x+3);
2. (x-4)(x+4); 3. (x-7)(x+7);
4. (x-8)(x+8); 5. (x-10)(x+10);
6.
(x-9)(x+9); 7. (x-5)(x+5);
8. (x-11)(x+11); 9. (x-12)(x+12);
10. (x-13)(x+13);
11.
(x-a)(x+a); 12. (x-b)(x+b);
13. (2x-3)(2x+3); 14. (3x-4)(3x+4);
15. (4x-7)(4x+7);
16.(5x-8)(5x+8);
17. (3x-10)(3x+10); 18. (7x-9)(7x+9);
19. (3x-5y)(3x+5y);
20.
(7x-11y)(7x+11y);
21. (5x-12a)(5x+12a); 22. (4x-9b)(4x+9b);
23. (3y-13x)(3y+13x);
24.
(6y-7x)(6y+7x);
25. 9(x-1)(x+1); 26. 3(x-2)(x+2);
27. 5(x-3)(x+3);
28.
4(x-4)(x+4); 29. 4(x-5)(x+5);
30. 8(x-3)(x+3); 31. 25(x-1)(x+1);
32. 7(x-10)(x+10);
33. x(x-4)(x+4);
34. 3x(x-5)(x+5); 35.
5x(x-4)(x+4); 36. 2x(x-5)(x+5);
37.
16x2(x-y)(x+y);
38. 9y2(y-x)(y+x);
39. 9y2(3y-x)(3y+x);
40.4x2(4x-y)(4x+y);
41.
12y2(y-x)(y+x);
42. 79y2(y-x)(y+x);
43. (x+2)2; 44. (x+3)2;
45. (x+1)2; 46. (x+6)2;
47. (x+7)2;
48. (x+10)2; 49. (x-4)2;
50. (x-2)2; 51. (x-5)2;
52. (x-6)2; 53. (x-9)2;
54. (x-12)2;
55.
5(x-2)2; 56. 2(x-5)2
; 57. 3(x+1)2; 58. 3(x-10)2
; 59. 6(x-3)2; 60. 4(x+4)2;
61. x(x+2)2 ;
62. x(x-6)2;
63. 9x(x-1)2; 64. 6x2(x-3)2
; 65. 12y2(y-2)2 ;
66. 2y2(y-7)2;
67. (x-2)(x+2)(x2+4);
68. (x-1)(x+1)(x2+1);
69. (x-3)(x+3)(x2+9);
70. (x-y)(x+y)(x2+y2)
;
71.
(3x-2y)(3x+2y)(9x2
+4y2); 72. (2x-3)(2x+3)(4x2+9);
73. (x2+9)(x2+1);
74. (x2+9)(x2+4);
75.
(x-3)(x+3)(x-2)(x+2);
76. (x-2)(x+2)(x-1)(x+1);
77. (x-5)(x+5)(x-2)(x+2);
78.
(y2+9)(y-2)(y+2);
79. (y-2)2(y+2)2;
80. (y-3)2(y+3)2.
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