The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.

1. Factoring Out GCF

2. To factor means to rewrite a quantity as a product (in a
nontrivial way).
Factoring Out GCF

3. To factor means to rewrite a quantity as a product (in a
nontrivial way). A quantity x that can’t be written as product
besides as 1*x is said to be prime.
Factoring Out GCF

4. To factor means to rewrite a quantity as a product (in a
nontrivial way). 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

5. Example A. Factor 12 completely.
To factor means to rewrite a quantity as a product (in a
nontrivial way). 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

6. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
To factor means to rewrite a quantity as a product (in a
nontrivial way). 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

7. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime, incomplete
factored completely
To factor means to rewrite a quantity as a product (in a
nontrivial way). 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

8. 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 (in a
nontrivial way). 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.

9. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5,
A common factor of two or more quantities is a factor
belongs to all the quantities.

10. 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 (in a
nontrivial way). 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. Sincer 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.

11. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor of 4ab, 6a are
A common factor of two or more quantities is a factor
belongs to all the quantities.

12. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor of 4ab, 6a are 2,
A common factor of two or more quantities is a factor
belongs to all the quantities.

13. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor of 4ab, 6a are 2, a,
A common factor of two or more quantities is a factor
belongs to all the quantities.

14. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor of 4ab, 6a are 2, a, 2a.
A common factor of two or more quantities is a factor
belongs to all the quantities.

15. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor 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.

16. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor 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.

17. 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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor 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.

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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor 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.

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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor 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.

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 (in a
nontrivial way). 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. Sincer 6 = 2*3, 15 = 3*5, 3 is a common factor.
b. The common factor of 4ab, 6a are 2, a, 2a.
c. The common factors of 6xy2, 15x2y2 are 3, x, y2, xy2, ..
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.

21. 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

22. 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}

23. 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.

24. 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}

25. 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.

26. 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}

27. 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.

28. 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} =

29. 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.

30. 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.

31. 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)

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
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.

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

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

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

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

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

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

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

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

41. (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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2

42. (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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1)

43. (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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

44. 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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

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

46. 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)
(the GCF is y)
(the GCF is 2a)
(the GCF is 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

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

48. 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) = (x + y)(a – 4)
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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

49. 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) = (x + y)(a – 4)
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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

50. 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) = (x + y)(a – 4)
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)
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

51. Factoring Out GCF
We sometimes pull out a negative sign from an expression.

52. Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
We sometimes pull out a negative sign from an expression.

53. Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
We sometimes pull out a negative sign from an expression.

54. Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

55. Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

56. Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

57. Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
= (2x – 5)(y – 1)
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

58. Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
= (2x – 5)(y – 1)
b. 3x – 3y + ax – ay
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

59. We sometimes pull out a negative sign from an expression.
Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
= (2x – 5)(y – 1)
b. 3x – 3y + ax – ay Group them into two groups.
= (3x – 3y) + (ax – ay)
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.

60. Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
= (2x – 5)(y – 1)
b. 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)
Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
We sometimes pull out a negative sign from an expression.

61. Factoring Out GCF
Example F. Pull out the negative sign from –2x + 5.
–2x + 5 = –(2x – 5)
Sometime we have to use factor twice.
Example G. Factor by pulling out twice.
a. y(2x – 5) – 2x + 5
= y(2x – 5) – (2x – 5)
= (2x – 5)(y – 1)
b. 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)
We sometimes pull out a negative sign from an expression.