The document discusses factoring quantities and finding common factors. It provides examples of factoring numbers like 12 and finding the common factors of expressions like 4ab and 6a. It defines the greatest common factor as the largest common factor of two or more quantities. It explains that factoring means writing a quantity as a product of other factors and that a prime number is one only divisible by 1 and itself.

1. Factoring Out GCF

2. There are three boxes A, B, and C as shown here.
Factoring Out GCF

3. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same.

4. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,

5. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,
or three bananas from each box,

6. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,
or three bananas from each box,
or two bananas and two carrot.

7. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,
or three bananas from each box,
or two bananas and two carrot.
A group of items which may be taken from each of the three
boxes is a group of common items.

8. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,
or three bananas from each box,
or two bananas and two carrot.
A group of items which may be taken from each of the three
boxes is a group of common items.
In this case the largest group of items which may be taken
from each of the three boxes consists of

9. There are three boxes A, B, and C as shown here.
Factoring Out GCF
We are to take items from all three boxes and the items taken
from the boxes must be the same. For example, we may take
two apples from each box,
or three bananas from each box,
or two bananas and two carrot.
A group of items which may be taken from each of the three
boxes is a group of common items.
In this case the largest group of items which may be taken
from each of the three boxes consists of
We define the “greatest common factor” in a similar way.

10. To factor means to rewrite a quantity as a product (without
using 1).
Factoring Out GCF

11. Example A. Factor 12 completely.
To factor means to rewrite a quantity as a product (without
using 1).
Factoring Out GCF

12. Example A. Factor 12 completely.
12 = 3 * 4
To factor means to rewrite a quantity as a product (without
using 1).
Factoring Out GCF

13. Example A. Factor 12 completely.
12 = 3 * 4
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime.
Factoring Out GCF

14. Example A. Factor 12 completely.
12 = 3 * 4
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime.
Factoring Out GCF
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

15. Example A. Factor 12 completely.
12 = 3 * 4
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

16. Example A. Factor 12 completely.
12 = 3 * 4
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
not prime
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

17. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
factored completelynot prime
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

18. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

19. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

20. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

21. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

22. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

23. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

24. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

25. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

26. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

27. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

28. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x, y2,
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

29. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

30. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..
The common factor may be a formula in in parenthesis:
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

31. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor means to rewrite a quantity as a product (without
using 1). A quantity x that can’t be written as product besides
as 1*x is said to be prime. To factor completely means each
factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factors of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..
The common factor may be a formula in in parenthesis:
d. The common factor of a(x+y), b(x+y) is (x+y).
A common factor of two or more quantities is a factor
belongs to all the quantities.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

32. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Similar to selecting fruits from
boxes, it’s the largest group of
common items (factors).

33. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36}

34. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} The GCF of a list of numbers
is the largest number that
may be divided by all the number.

35. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12. The GCF of a list of numbers
is the largest number that
may be divided by all the number.

36. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a}
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

37. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

38. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2}
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

39. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

40. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
d. GCF{x3y5, x4y6, x5y4} =
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

41. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
d. GCF{x3y5, x4y6, x5y4} = x3y4.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

42. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
d. GCF{x3y5, x4y6, x5y4} = x3y4.
The Extraction Law
Distributive law interpreted backward gives the Extraction Law,
that is, common factors may be extracted from sums or
differences.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

43. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
d. GCF{x3y5, x4y6, x5y4} = x3y4.
The Extraction Law
Distributive law interpreted backward gives the Extraction Law,
that is, common factors may be extracted from sums or
differences.
AB ± AC  A(B±C)
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

44. The greatest common factor (GCF) is the common factor
that has the largest coefficient and highest degree of each
factor among all common factors.
Factoring Out GCF
Example C. Find the GCF of the given quantities.
a. GCF{24, 36} = 12.
b. GCF{4ab, 6a} = 2a.
c. GCF {6xy2, 15 x2y2} = 3xy2.
d. GCF{x3y5, x4y6, x5y4} = x3y4.
The Extraction Law
Distributive law interpreted backward gives the Extraction Law,
that is, common factors may be extracted from sums or
differences.
AB ± AC  A(B±C)
This procedure is also called “factoring out a common factor”.
To factor, the first step always is to factor out the GCF,
then factor the “left over” if it’s needed.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

45. Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y

46. (the GCF is y)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y

47. (the GCF is y)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y

48. (the GCF is y)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a

49. (the GCF is y)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3)

50. (the GCF is y)
(the GCF is 2a)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3)

51. (the GCF is y)
(the GCF is 2a)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)

52. (the GCF is y)
(the GCF is 2a)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2

53. (the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)

54. (the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Factoring Out GCF
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

55. Factoring Out GCF
We may pull out common factors that are ( )'s.
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

56. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

57. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

58. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

59. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

60. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3) (3x – 2)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

61. Factoring Out GCF
We may pull out common factors that are ( )'s.
Example E. Factor
a. a(x + y) – 4(x + y)
Pull out the common factor (x + y)
a(x + y) – 4(x + y) = (a – 4)(x + y)
b. Factor (2x – 3)3x – 2(2x – 3)
Pull out the common factor (2x – 3),
(2x – 3)3x – 2(2x – 3) = (2x – 3) (3x – 2)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)
Note the order
of the “( )’s”
doesn’t matter
because AB=BA.

62. Factor by Grouping
There are special four–term formulas where we have to
separate the terms into two pairs,

63. There are special four–term formulas where we have to
separate the terms into two pairs,
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay
Factor by Grouping

64. There are special four–term formulas where we have to
separate the terms into two pairs,
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay)
Factor by Grouping

65. There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay)
Factor by Grouping

66. There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y)
Factor by Grouping

67. There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y)
Factor by Grouping
A common parenthesis–factor

68. There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y) Pull the factor (x – y) again.
= (3 + a)(x – y)
Factor by Grouping

69. We may need to pull out the negative sign
e.g. writing –4x + 10 as –(2x – 5),
in the expression to reveal the common factor.
b. y(2x – 5) – 4x + 10
There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y) Pull the factor (x – y) again.
= (3 + a)(x – y)
Factor by Grouping

70. We may need to pull out the negative sign
e.g. writing –4x + 10 as –(2x – 5),
in the expression to reveal the common factor.
b. y(2x – 5) – 4x + 10
= y(2x – 5) – 2(2x – 5)
There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y) Pull the factor (x – y) again.
= (3 + a)(x – y)
Factor by Grouping

71. We may need to pull out the negative sign
e.g. writing –4x + 10 as –(2x – 5),
in the expression to reveal the common factor.
b. y(2x – 5) – 4x + 10
= y(2x – 5) – 2(2x – 5)
= (y – 2) (2x – 5)
There are special four–term formulas where we have to
separate the terms into two pairs, factor out each pair’s GCF
to reveal a common parenthesis–factor, then we pull out this
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay) Factor out the GCF of each group.
= 3(x – y) + a(x – y) Pull the factor (x – y) again.
= (3 + a)(x – y)
Factor by Grouping

72. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
Factor by Grouping

73. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
Factor by Grouping

74. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
Factor by Grouping

75. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Factor by Grouping

76. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Factor by Grouping

77. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Group x2 – 2x – x + 2 into two groups.
x2 – 2x – x + 2
Factor by Grouping

78. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Group x2 – 2x – x + 2 into two groups.
x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2)
Factor by Grouping

79. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Group x2 – 2x – x + 2 into two groups.
x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2) Factor out the GCF of each group.
= x(x – 2) –1(x – 2)
Factor by Grouping

80. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Group x2 – 2x – x + 2 into two groups.
x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2) Factor out the GCF of each group.
= x(x – 2) –1(x – 2) Pull the factor (x – 2) again.
= (x – 2 )(x – 1)
Factor by Grouping

81. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
The product of two binomials is a trinomials:
(#x + #)(#x + #)  ax2 + bx + c
For example (x + 2)(2x + 3) = 2x2 + 7x + 6.
To factor a trinomial means to a undo the multiplication and
write the trinomial as a product of two binomials, if possible.
ax2 + bx + c  (#x + #)(#x + #)
Example G.
Factor the the trinomial x2 – 3x + 2 by grouping given that
x2 – 3x + 2 = x2 – 2x – x + 2
Group x2 – 2x – x + 2 into two groups.
x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2) Factor out the GCF of each group.
= x(x – 2) –1(x – 2) Pull the factor (x – 2) again.
= (x – 2 )(x – 1) We will use grouping as the default
method for factoring trinomials.
Factor by Grouping

82. Factoring Out GCF
Exercise. A. Find the GCF of the listed quantities.
Factoring Out GCF
1. {4, 6 } 2. {12, 18 } 3. {32, 20, 12 } 4. {25, 20, 30 }
5. {4x, 6x2 } 6. {12x2y, 18xy2 }
7. {32A2B3, 20A3B3, 12 A2B2}
8. {25x7y6z6, 20y7z5x6, 30z8x7y6 }
B. Factor out the GCF.
9. 4 – 6y 10. 12x + 18y 11. 32A + 20B – 12C
12. 25x + 20y – 30 13. –4x + 6x2
14. –12x2y – 18xy2 15. 32A2B3 – 20A3B3 – 12A2B2}
16. 25x7y6z6 – 20y7z5x6 + 30z8x7y6
17. 4x4 – 8x3 + 2x2 18. 20x4 – 5x2
19. x(x – 2) + 3(x – 2) 20. 4x(2x – 3) – 5(2x – 3)
C. Factor out the “–”.
21. –2y + 4 22. –3x + 18 23. –5x + 15 24. –8x + 16

83. Factoring Out GCF
D. Factor, use grouping if it’s necessary.
25. y2 – 2y + 3y – 6 26. x2 + 3x + 6x + 18
27. y2 – 2y – 3y + 6 28. x2 + 3x – 6x – 18
29. y2 – y + 4y – 4 30. x2 – 5x – 2x + 10
31. 2y2 – y – 6y + 3 32. 3x2 + 2x – 6x – 4
33. 4x2 + 6x – 6x – 9 34. –3x2 + 4x – 6x + 8
35. –5y2 + 10y – 3y + 6 36. –x2 + 3x – 7x + 21
37. 2y2 – xy – 6xy + 3x2 38. 3x2 + 2xy – 6xy – 4y2
39. –5x2 + 2xy – 20xy + 8y2 40. –14x2 + 21xy – 8xy + 12y2

84. 35. –3x3 – 30x2 – 48x34. –yx2 + 4yx + 5y
36. –2x3 + 20x2 – 24x
40. 4x2 – 44xy + 96y2
37. –x2 + 11xy + 24y2
38. x4 – 6x3 + 36x2 39. –x2 + 9xy + 36y2
C. Factor. Factor out the GCF, the “–”, and arrange the
terms in order first.
D. Factor. If not possible, state so.
41. x2 + 1 42. x2 + 4 43. x2 + 9 43. 4x2 + 25
44. What can you conclude from 41–43?
Factoring Trinomials and Making Lists

85. Exercise. A. Factor. If it’s prime, state so.
1. x2 – x – 2 2. x2 + x – 2 3. x2 – x – 6 4. x2 + x – 6
5. x2 – x + 2 6. x2 + 2x – 3 7. x2 + 2x – 8 8. x2 – 3x – 4
9. x2 + 5x + 6 10. x2 + 5x – 6
13. x2 – x – 20
11. x2 – 5x – 6
12. x2 – 5x + 6
17. x2 – 10x – 24
14. x2 – 8x – 20
15. x2 – 9x – 20 16. x2 – 9x + 20
18. x2 – 10x + 24 19. x2 – 11x + 24 20. x2 – 11x – 24
21. x2 – 12x – 36 22. x2 – 12x + 36 23. x2 – 13x – 36
24. x2 – 13x + 36
B. Factor. Factor out the GCF, the “–”, and arrange the
terms in order first if necessary.
29. 3x2 – 30x – 7227. –x2 – 5x + 14 28. 2x3 – 18x2 + 40x
30. –2x3 + 20x2 – 24x
25. x2 – 36 26. x2 + 36
31. –2x4 + 18x2
32. –3x – 24x3 + 22x2 33. 5x4 + 10x5 – 5x3
Factoring Trinomials and Making Lists