CHAPTER 4 Factoring
Chapter 4
Factoring
Section 4.1:
Greatest Common Factor and Factoring by
Grouping
 GCF and Grouping
GCF and Grouping
Finding the Greatest Common Factor:
Example:
Solution:
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University of Houston Department of Mathematics
SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Factoring Out the Greatest Common Factor:
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CHAPTER 4 Factoring
Example:
Solution:
Factoring by Grouping:
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SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Additional Example 1:
Solution:
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CHAPTER 4 Factoring
Additional Example 2:
Solution:
The GCF is the product of the factors that are shared by all three monomials.
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SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Additional Example 3:
Solution:
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CHAPTER 4 Factoring
Additional Example 4:
Solution:
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SECTION 4.1 Greatest Common Factor and Factoring by Grouping
Additional Example 5:
Solution:
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Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
Find the GCF (Greatest Common Factor) of the
following monomials.
1.
18x3 y 2 , 24x2 y5 , 12xy 4
2.
20 x3 y5 , 32 x7 y3 , 8x4 y9
7 4
4 5
29. (a) xy  5 y
a b , 7a b , 14a b
4.
12c6 d10 , 15c4 d , 21c7 d 4
5.
16a3b12c10 , 32a5bc6 , 100a7 c5
6.
7 3
28. 4x7 y 4 z  35 y6 z 4  9x2 y6 z3
Factor the following expressions.
7 8
3.
5 2
27. 10a3b2 c5  21a 4 c7  49b3c8
(b) x  x  4  5  x  4
30. (a) xy  3 y
(b) x  x  6  3 x  6
7 3
30a b , 90a c , 45b c
7.
10x y z , 18x y z , 7 x y z
31. (a) 3b  ab
(b) 3  c  5  a  c  5
8.
9 x7 y5 z3 , 50x4 y8 , 20x6 y5 z 2
32. (a) ap  cp
6 9 8
3 5 7
5 6 4
Find the GCF of the terms of the polynomial and
factor it out. If the leading coefficient is negative, then
factor out the negative of the GCF.
9.
5a  10
(b) a  b  2  c b  2
33. 3a(a  5)  4b(a  5)
34. 4 x( x  7)  3 y( x  7)
35. 2 x( x  8)  ( x  8)
10. 4 x  12
36. 3b(b  2)  (b  2)
11. 3b  15
37. ( x  3)( x  5)  ( x  2)( x  5)
12. 4 y  24
38. ( x  4)( x  1)  ( x  4)( x  6)
13. 9x  24 y
39. (a  2)(4a  3)  (a  8)(a  2)
14. 10a  25c
40. (3a  1)(2a  6)  (3a  1)(a  2)
15. 6x  8xy
16. 8ab  12bc
Factor by grouping.
17. 6a3b2  2ab
41. 2b  2c  ab  ac
18. 3ac  6a5c7
42. 3x  3 y  xz  yz
19. 15r 2t  20rt 2
43. 5 y  5z  xy  xz
20. 30u 4 v3  2u 3 w6
44. 4a  4b  ca  cb
21. 4 x3  2 x2  8x
45. x2  3x  xy  3 y
22. 18x5  36 x3  45x2
46. xy  3x  5 y 15
23. 5x3 y 2  3x4 y5  7 x8 y3
24. 20a3b6  8ab4  12a5b2
25. 35a7b4 c9  28a 2b5c  21a3b9c6
26. 36x3 y7 z8  12x2 y5 z 4  48x2 y6 z 7
47. ac  ad  bc  bd
48. p2  pr  tp  tr
49. xy  4x  y  4
50. b  2  ab  2a
51. y 2  xy  y  x
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Exercise Set 4.1: Greatest Common Factor and Factoring by Grouping
52. px  p2  x  p
53. 12b  8  3ab  2a
54. 18xy  24x  15 y  20
55. 6t 2  2tx  3t  x
56. 15x2  5xy  6x  2 y
57. 12ac  3ad  8bc  2bd
58. 24xy 15xz  8 y  5z
Factor by grouping. (Hint: Use groups of three.)
59. ad  ae  af  bd  be  bf
60. xy  xz  4x  3y  3z  12
61. 3x2  12xz 15x  2xy  8 yz  10 y
62. 12ab  8ac  20ad  3b2  2bc  5bd
Each of the following expressions contains like terms.
Do not combine the like terms; instead, simply factor
by grouping. (This method will be helpful in the next
section when factoring trinomials.)
63. x 2  3x  2 x  6
64. x 2  5x  7 x  35
65. x2  4 x  3x  12
66. x2  3x  6 x  18
67. 6 x2  10 x  9 x  15
68. 21x2  3x  14 x  2
69. 9 x2  21x  6 x  14
70. 25x2  5x  20 x  4
71. 4 x2  14 x  14 x  49
72. 9 x 2  15 x  15x  25
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CHAPTER 4 Factoring
Section 4.2:
Factoring Special Binomials and Trinomials
 Special Factor Patterns
Special Factor Patterns
Factoring the Difference of Two Squares:
Example:
Solution:
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SECTION 4.2 Factoring Special Binomials and Trinomials
Note:
Factoring the Difference of Two Cubes:
Example:
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CHAPTER 4 Factoring
Solution:
Factoring the Sum of Two Cubes:
Example:
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SECTION 4.2 Factoring Special Binomials and Trinomials
Solution:
Factoring Perfect Square Trinomials:
Example:
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CHAPTER 4 Factoring
Solution:
Additional Example 1:
Solution:
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SECTION 4.2 Factoring Special Binomials and Trinomials
Additional Example 2:
Solution:
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CHAPTER 4 Factoring
(c) The monomials 10 yz 3 and 10 y share a common factor of 10 y. The first step in
factoring the given binomial to factor out the GCF of 10 y .
Additional Example 3:
Solution:
Additional Example 4:
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University of Houston Department of Mathematics
SECTION 4.2 Factoring Special Binomials and Trinomials
Solution:
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Exercise Set 4.2: Factoring Special Binomials and Trinomials
17. p 2  1
Multiply the following.
1.
(a)  x  4 x  4
(b)
(c)
2.
(a)
(b)
(c)
18. 1  p 2
 x  4
 x  4 2
2
19. x2  100
20. x2  4
 x  9 x  9
 x  9 2
 x  9 2
21. 25  c2
22. 144  d 2
23. 4b 2  9
24. 25x2  49
Answer True or False.
25. 16 x 2  1
3.
x 2  49   x  7 
4.
x 2  64   x  8 x  8
5.
 x  6 2  x 2  12 x  36
6.
 x  4
7.
 x  10 x  10  x2  100
8.
x 2  24 x  144   x  12 
9.
x  81   x  9 
10.
 x  5 2  x 2  25
2
2
26. 36 x 2  1
27. 49 x2  100 y 2
28. 64a 2  25b2
 x  16
2
2
2
2
Factor the following polynomials. If the polynomial
can not be factored any further within the real
number system, then write the original polynomial as
your answer.
11. (a) x 2  9
29. 25c 2  16d 2
30. 4 z 2  9w2
31.
x2
4
9
32.
x2
1
16
33.
x2 a2

y 2 b2
34.
p2 r 2

q2 t 2
35.
16 x2 y 2

25
9
36.
a 2b2 100
 2
81
c
(b) x 2  9
(c) x  6 x  9
2
(d) x 2  6 x  9
12. (a) x 2  25
(b) x 2  25
(c) x  10 x  25
37. x2  20 x  100
2
(d) x 2  10 x  25
38. x 2  8 x  16
39. x 2  2 x  1
13. x 2  49
14. x  36
40. x 2  14 x  49
2
15. x  144
41. x2  18x  81
2
16. a  81
42. x2  24 x  144
2
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43. 4 x 2  12 x  9
University of Houston Department of Mathematics
Exercise Set 4.2: Factoring Special Binomials and Trinomials
44. 9 x2  30 x  25
45. 25x2  40 x  16
46. 36 x 2  12 x  1
47. x2  2bx  b2
48. x 2  2cx  c 2
49. 4b2 c2  20bcd  25d 2
50. 9x2  24xyz  16 y 2 z 2
When the remainder is zero, the dividend can be
written as a product of two factors (the divisor and the
quotient), as shown below.
30
 6 , so 30  5  6 .
5
x2  x  6
 x  2 , so x 2  x  6   x  3  x  2 
x3
In the following examples, use either long division or
synthetic division to find the quotient, and then write
the dividend as a product of two factors.
51.
x3  8
x2
52.
x3  27
x3
Factor the following polynomials.
53. x3  64
54. m3  1
55. p3  27
56. x3  125
57. x3  y3
58. c3  d 3
59. 125a3  8b3
60. 64x3  27 y3
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CHAPTER 4 Factoring
Section 4.3:
Factoring Polynomials
 Techniques for Factoring Trinomials
Techniques for Factoring Trinomials
Factorability Test for Trinomials:
Example:
Solution:
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SECTION 4.3 Factoring Polynomials
Factoring Trinomials with Leading Coefficient 1:
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CHAPTER 4 Factoring
Example:
Solution:
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SECTION 4.3 Factoring Polynomials
Factoring Trinomials with Leading Coefficient Different from 1:
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CHAPTER 4 Factoring
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SECTION 4.3 Factoring Polynomials
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CHAPTER 4 Factoring
Example:
Solution:
Additional Example 1:
(a) 2 x 2  3 x  8
(b) 42 x 2  25 x  3
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SECTION 4.3 Factoring Polynomials
Solution:
Additional Example 2:
Solution:
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CHAPTER 4 Factoring
Additional Example 3:
Solution:
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SECTION 4.3 Factoring Polynomials
Additional Example 4:
Solution:
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CHAPTER 4 Factoring
Additional Example 5:
Solution:
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University of Houston Department of Mathematics
Exercise Set 4.3: Factoring Polynomials
At times, it can be difficult to tell whether or not a
quadratic of the form ax 2  bx  c can be factored
into the form  dx  e  fx  g  , where a, b, c, d, e, f,
and g are integers. If b2  4ac is a perfect square, then
the quadratic can be factored in the above manner.
For each of the following problems,
(a) Compute b2  4ac .
(b) Use the information from part (a) to
determine whether or not the quadratic can
be written as factors with integer coefficients.
(Do not factor; simply answer Yes or No.)
21. x 2  16 x  64
22. x 2  6 x  9
23. x2  15x  56
24. x2  6 x  27
25. x2  11x  60
26. x 2  19 x  48
27. x 2  17 x  42
28. x 2  12 x  64
1.
x2  5x  3
29. x 2  49
2.
x 2  7 x  10
30. x 2  36
3.
x 2  6 x  16
31. x 2  3
4.
x2  6 x  4
32. x 2  8
5.
9  x2
33. 9 x2  25
6.
7x  x 2
34. 16 x2  81
7.
2 x2  7 x  4
35. 2 x 2  5x  3
8.
6 x2  x  1
36. 3x 2  16 x  15
9.
2 x2  2 x  5
37. 8x 2  2 x  3
10. 5 x 2  4 x  1
38. 4 x2  16 x  15
39. 9 x2  9 x  4
Factor the following polynomials. If the polynomial
can not be rewritten as factors with integer
coefficients, then write the original polynomial as your
answer.
40. 5 x 2  17 x  6
41. 4 x2  3x  10
11. x 2  4 x  5
42. 9 x2  21x  10
12. x 2  9 x  14
43. 12 x 2  17 x  6
13. x2  5x  6
44. 8x 2  26 x  7
14. x 2  x  6
15. x 2  7 x  12
16. x 2  8 x  15
17. x 2  12 x  20
18. x  7 x  18
Factor the following. Remember to first factor out the
Greatest Common Factor (GCF) of the terms of the
polynomial, and to factor out a negative if the leading
coefficient is negative.
45. x 2  9 x
2
19. x  5x  24
46. x2  16 x
2
20. x  9 x  36
47. 5x2  20 x
2
MATH 1300 Fundamentals of Mathematics
48. 4 x2  28x
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Exercise Set 4.3: Factoring Polynomials
49. 2 x2  18
50. 8 x 2  8
51. 5x4  20 x2
52. 3x3  75 x
53. 2 x 2  10 x  8
54. 3x2  12 x  63
55. 10 x 2  10 x  420
56. 4 x2  40 x  100
57. x3  9 x2  22 x
58. x3  7 x 2  6 x
59.  x3  4 x 2  4 x
60. x5  10 x 4  21x3
61. x4  6 x3  6 x2
62.  x3  2 x2  80 x
63. 9 x5  100 x3
64. 49 x12  64 x10
65. 50 x 2  55 x  15
66. 30 x2  24 x  72
Factor the following polynomials. (Hint: Factor first
by grouping, and then continue to factor if possible.)
67. x3  2 x 2  25x  50
68. x3  3x 2  4 x  12
69. x3  5x2  4 x  20
70. 9 x3  18x 2  25x  50
71. 4 x3  36 x2  x  9
72. 9 x3  27 x 2  4 x  12
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SECTION 4.4 Using Factoring to Solve Equations
Section 4.4:
Using Factoring to Solve Equations
 Solving Quadratic Equations by Factoring
 Solving Other Polynomials Equations by Factoring
Solving Quadratic Equations by Factoring
Zero-Product Property:
Example:
Solution:
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CHAPTER 4 Factoring
Example:
Solution:
The x-Intercepts of the Graph of a Quadratic Function:
Example:
Solution:
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SECTION 4.4 Using Factoring to Solve Equations
Additional Example 1:
Solution:
Additional Example 2:
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CHAPTER 4 Factoring
Solution:
Additional Example 3:
Solution:
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SECTION 4.4 Using Factoring to Solve Equations
Additional Example 4:
Solution:
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CHAPTER 4 Factoring
Additional Example 5:
Solution:
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SECTION 4.4 Using Factoring to Solve Equations
Additional Example 6:
Solution:
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CHAPTER 4 Factoring
(c) Since a  1  0 , the parabola opens upward.
Solving Other Polynomial Equations by Factoring
Solving Polynomial Equations by Factoring:
Example:
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University of Houston Department of Mathematics
SECTION 4.4 Using Factoring to Solve Equations
Solution:
Example:
Solution:
The x-Intercepts of the Graph of a Polynomial Function:
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CHAPTER 4 Factoring
Example:
Solution:
292
University of Houston Department of Mathematics
SECTION 4.4 Using Factoring to Solve Equations
Additional Example 1:
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
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CHAPTER 4 Factoring
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Exercise Set 4.4: Using Factoring to Solve Equations
Solve the following equations by factoring.
28.  4 x 2  20 x  24  0
1.
x 2  10 x  21  0
29. 80 x 2  230 x  30  0
2.
x 2  13 x  40  0
30. 12 x 2  75 x  18  0
3.
x 2  8 x  12  0
31. x3  5x 2  6 x  0
4.
x 2  6 x  40  0
32. x3  7 x2  18x  0
5.
x( x  2)  35
6.
x( x  8)  20
Each of the quadratic functions below is written in the
form f ( x )  ax 2  bx  c . The graph of a quadratic
7.
x 2  14 x  72
function is a parabola with vertex, where h   2ba
8.
x 2  60  11x
and k  f
9.
2 x 2  7 x  15  0
10. 3x 2  7 x  4  0
11. 6 x 2  17 x  12
12. 10 x 2  7 x  6
13. 3x 2  5x  2
14. 8x 2  6 x  5  0
15. x 2  25  0
16. x 2  49  0
17. 4 x 2  9  0
18. 36 x 2  25  0
Solve the following equations by factoring. To simplify
the process, remember to first factor out the Greatest
Common Factor (GCF) and to factor out a negative if
the leading coefficient is negative.
  2ba  .
(a) Find the x-intercept(s) of the parabola by
setting f ( x)  0 and solving for x.
(b) Write the coordinates of the x-intercept(s)
found in part (a).
(c) Find the y-intercept of the parabola and write
its coordinates.
(d) Give the coordinates of the vertex (h, k) of the
parabola, using the formulas h   2ba and
k f
  2ba  .
(e) Does the parabola open upward (with the
vertex being the lowest point on the graph) or
downward (with the vertex being the highest
point on the graph)?
(f) Find the axis of symmetry. (Be sure to write
your answer as an equation of a line.)
(g) Draw a graph of the parabola that includes
the features from parts (b) through (e).
33. f ( x)  x2  6x  8
19. x 2  8 x  0
34. f ( x)  x2  2 x  15
20. x 2  10 x  0
35. f ( x)  x2  8x  16
21.  x2  5x  36  0
36. f ( x)   x2  10x 16
22.  x 2  14 x  48  0
37. f ( x)   x2  4 x  21
23.  3x 2  21x  0
38. f ( x)  x2  10x  25
24. 5x 2  30 x  0
39. f ( x)  3x2  12x  36
25. 3x 2  12  0
26.  7 x 2  7  0
27.  5x  15 x  90  0
40. f ( x)  4 x2  8x  5
41. f ( x)  x2 16
2
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Exercise Set 4.4: Using Factoring to Solve Equations
42. f ( x)  25  x2
43. f ( x)  9  4x2
44. f ( x)  9 x2  100
Find the x-intercept(s) of the following.
45. f ( x)  x3  7 x2  10x
46. f ( x)  x3  2x2  99x
47. f ( x)  x3  25x
48. f ( x)  x3  4 x
49. f ( x)  x3  2 x2  9 x  18
50. f ( x)  x3  4 x2  x  4
For each of the following problems:
(a) Model the situation by writing appropriate
equation(s).
(b) Solve the equation(s) and then answer the
question posed in the problem.
51. The length of a rectangular frame is 5 cm longer
than its width. If the area of the frame is 36 cm2,
find the length and width of the frame.
52. The width of a rectangular garden is 8 m shorter
than its length. If the area of the field is 180 m2,
find the length and the width of the garden
53. The height of a triangle is 3 cm shorter than its
base. If the area of the triangle is 90 cm2, find the
base and height of the triangle.
54. Find x if the area of the figure below is 26cm2.
(Note that the figure may not be drawn to scale.)
x cm
3 cm
8 cm
x cm
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University of Houston Department of Mathematics