The document discusses factoring quantities and finding common factors and greatest common factors (GCF). It provides examples of factoring numbers like 12 and finding the common factors and GCF of expressions. Factoring means rewriting a quantity as a product without 1. A common factor is one that belongs to all quantities, and the GCF is the largest common factor considering coefficients and degrees of factors. Examples show finding the common factors of expressions and the GCF of quantities like 24 and 36.

1. Factoring Out GCF

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

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

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

5. 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

6. 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.

7. 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.

8. 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.

9. 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.

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 (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.

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 (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.

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 (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.

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 (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.

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 (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.

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 (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.

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 (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.

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 (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.

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
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.

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, 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.

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.
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.

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 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.

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, 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.
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

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

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}

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.

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}

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.

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}

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.

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} =

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.

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.

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} = 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)

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} = 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.

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

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

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

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

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

41. (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)

42. (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

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)

44. (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)

45. 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)

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)
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)
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) = (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)
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) = (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)
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) = (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)
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) = (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)
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 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)
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.

52. Factoring Out GCF
There are special four–term formulas where we have to
separate the terms into two pairs,

53. Factoring Out GCF
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

54. Factoring Out GCF
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)

55. Factoring Out GCF
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)

56. Factoring Out GCF
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)

57. Factoring Out GCF
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. x(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)

58. Factoring Out GCF
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. x(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)

59. Factoring Out GCF
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. x(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)

60. Trinomials (three-term) are polynomials of the form
ax2 + bx + c where a, b, and c are numbers.
Factoring Trinomials and Making Lists

61. 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
Factoring Trinomials and Making Lists

62. Factoring Trinomials and Making Lists
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
Hence, to factor a trinomial, we write the trinomial as a
product of two binomials, if possible,

63. 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
Hence, to factor a trinomial, we write the trinomial as a
product of two binomials, if possible, that is:
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists

64. 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
Hence, to factor a trinomial, we write the trinomial as a
product of two binomials, if possible, that is:
ax2 + bx + c  (#x + #)(#x + #)
We start with the case a = 1, trinomials of the form x2 + bx + c.
Factoring Trinomials and Making Lists

65. 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
Hence to “factor a trinomial” means, if possible,
to write the trinomial as a product of two binomials, that is:
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists

66. Example A.
a. Factor x2 + 5x + 6
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c

67. Example A.
a. Factor x2 + 5x + 6
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.

68. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.

69. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.
To carry this out, make a list of all the possible u and v
such that uv = c,

70. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
List all such u and v where uv = 6:
(1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) = 6
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.
To carry this out, make a list of all the possible u and v
such that uv = c,

71. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
List all such u and v where uv = 6:
(1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) = 6
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.
To carry this out, make a list of all the possible u and v
such that uv = c, then search for the pair that satisfies u + v = b.
Such a pair of u and v may or may not exist.

72. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
List all such u and v where uv = 6:
(1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) = 6
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.
To carry this out, make a list of all the possible u and v
such that uv = c, then search for the pair that satisfies u + v = b.
Such a pair of u and v may or may not exist.
2, 3 is the pair where u + v = 5.

73. Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6,
so we need u and v where uv = 6 and u + v = 5.
List all such u and v where uv = 6:
(1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) = 6
so x2 + 5x + 6 = (x + 2)(x + 3).
Factoring Trinomials and Making Lists
Factoring the trinomial x2 + bx + c
To factor the trinomial x2 + bx + c, search for a pair of numbers
u and v such that uv = c, and u + v = b.
To carry this out, make a list of all the possible u and v
such that uv = c, then search for the pair that satisfies u + v = b.
Such a pair of u and v may or may not exist.
2, 3 is the pair where u + v = 5.

74. c. Factor x2 + 5x – 6
We want (x + u)(x + v) = x2 + 5x – 6, so we need uv = –6 and
u + v = 5.
Since -6 = (–1)(6) = (1)(–6) = (–2)(3) =(2)(–3) and –1 + 6 = 5,
so x2 + 5x – 6 = (x – 1)(x + 6).
b. Factor x2 – 5x + 6
We want (x + u)(x + v) = x2 – 5x + 6, so we need u and v
where uv = 6 and u + v = –5.
Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and –2 – 3 = –5,
so x2 – 5x + 6 = (x – 2)(x – 3).
Example A.
a. Factor x2 + 5x + 6
We want (x + u)(x + v) = x2 + 5x + 6, so we need u and v
where uv = 6 and u + v = 5.
Since 6 = (1)(6) = (2)(3) = (-1)(-6) = (-2)(-3) and 2x + 3x = 5x,
so x2 + 5x + 6 = (x + 2)(x + 3)
Factoring Trinomials and Making Lists

75. Observations About Signs
Factoring Trinomials and Making Lists

76. Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
Factoring Trinomials and Making Lists

77. Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
Factoring Trinomials and Making Lists

78. Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
Factoring Trinomials and Making Lists

79. Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
if b is negative, then both are negative.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
Factoring Trinomials and Making Lists

80. {
Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
if b is negative, then both are negative.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
x2 – 5x + 6 = (x – 2)(x – 3)
Factoring Trinomials and Making Lists

81. {
Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
if b is negative, then both are negative.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
x2 – 5x + 6 = (x – 2)(x – 3)
2. If c is negative, then u and v have opposite signs.
Factoring Trinomials and Making Lists

82. {
Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
if b is negative, then both are negative.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
x2 – 5x + 6 = (x – 2)(x – 3)
2. If c is negative, then u and v have opposite signs. The
one with larger absolute value has the same sign as b.
Factoring Trinomials and Making Lists

83. {
Observations About Signs
Given that x2 + bx + c = (x + u)(x + v) so that uv = c,
we observe the following.
1. If c is positive, then u and v have same sign.
In particular,
if b is also positive, then both are positive.
if b is negative, then both are negative.
From the examples above
x2 + 5x + 6 = (x + 2)(x + 3)
x2 – 5x + 6 = (x – 2)(x – 3)
2. If c is negative, then u and v have opposite signs. The
one with larger absolute value has the same sign as b.
From the example above
x2 – 5x – 6 = (x – 6)(x + 1)
Factoring Trinomials and Making Lists

84. Example B.
a. Factor x2 + 4x – 12
We need u and v having opposite signs such that uv = –12,
u + v = +4. Since -12 = (-1)(12) = (-2)(6) = (-3)(4)…
They must be –2 and 6 hence x2 + 4x – 12 = (x – 2)(x + 6).
b. Factor x2 – 8x – 12
We need u and v such that uv = –12, u + v = –8 with
u and v having opposite signs. This is impossible.
Hence x2 – 8x – 12 is prime.
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

86. 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

87. 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

88. 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