Factoring

1. Factoring Trinomials and Making Lists

2. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
Factoring Trinomials and Making Lists

3. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
Factoring Trinomials and Making Lists

4. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Factoring Trinomials and Making Lists

5. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials,
Factoring Trinomials and Making Lists

6. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists

7. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:

8. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,

9. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)

10. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)
ll. the ones that are prime or no factorable, such as
x2 + 2x + 3

11. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)
ll. the ones that are prime or no factorable, such as
x2 + 2x + 3  (#x + #)(#x + #) (Not possible!)

12. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)
ll. the ones that are prime or no factorable, such as
x2 + 2x + 3  (#x + #)(#x + #)
Our jobs are to determine which trinomials:
(Not possible!)

13. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)
ll. the ones that are prime or no factorable, such as
x2 + 2x + 3  (#x + #)(#x + #)
Our jobs are to determine which trinomials:
1. are factorable and factor them,
(Not possible!)

14. For our discussions, trinomials (three-term) in x are polynomials
of the form ax2 + bx + c where a (≠ 0), b, and c are numbers.
We obtain trinomials from (#x + #)(#x + #)  ax2 + bx + c.
For example, (x + 2)(x + 1)  x2 + 3x + 2,
the trinomial with a = 1, b = 3, and c = 2.
Hence, "to factor a trinomial" means to convert the trinomial
back as a product of two binomials, that is,
ax2 + bx + c  (#x + #)(#x + #)
Factoring Trinomials and Making Lists
The Basic Fact About Factoring Trinomials:
There are two types of trinomials,
l. the ones that are factorable such as
x2 + 3x + 2  (x + 2)(x + 1)
ll. the ones that are prime or no factorable, such as
x2 + 2x + 3  (#x + #)(#x + #)
Our jobs are to determine which trinomials:
1. are factorable and factor them,
2. are prime so we won’t waste time on trying to factor them.
(Not possible!)

15. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.

16. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".

17. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.

18. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,

19. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,
I II
12 12
97
vu vu

20. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,
I II
12 12
97
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu

21. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,
I II
12 12
97
121 121
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu

22. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,
I II
12 12
97
121
62
121
62
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu

23. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
and if possible,
I II
12 12
97
121
62
43
121
62
43
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu

24. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
I II
12 12
97
121
62
43
121
62
43
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu
For table I. we see that
3 and 4 fit the conditions,
i.e. 3*4 = 12 and 3 + 4 = 7.
and if possible,

25. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
I II
12 12
97
121
62
43
121
62
43
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu
For table I. we see that
3 and 4 fit the conditions,
i.e. 3*4 = 12 and 3 + 4 = 7.
For table II. it’s not possible
to have that u*v = 12 and that u + v = 9.
and if possible,

26. Factoring Trinomials and Making Lists
One way to identify which is which is by making lists.
A list is a record of all the possibilities following some criteria
such as the list of "all my cousins".
The lists we will make are lists of numbers.
Example A.
Given the following X-table,
find two numbers u and v
such that:
i. u*v is the top number
ii. u + v is the bottom number
Not possible!
I II
12 12
97
121
62
43
121
62
43
Let’s list all the u’s and v’s
such that u*v=12
in an orderly fashion.
vu vu
For table I. we see that
3 and 4 fit the conditions,
i.e. 3*4 = 12 and 3 + 4 = 7.
For table II. it’s not possible
to have that u*v = 12 and that u + v = 9.
and if possible,

27. The ac-Method (for factoring trinomial)
Factoring Trinomials and Making Lists

28. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
Factoring Trinomials and Making Lists

29. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Factoring Trinomials and Making Lists

30. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Example B. Factor x2 – x – 6 by grouping.
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

31. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

32. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6 put the four terms into two pairs
= (x2 – 3x) + (2x – 6)
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

33. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6 put the four terms into two pairs
= (x2 – 3x) + (2x – 6) take out the GCF of each pair
= x(x – 3) + 2(x – 3) take out the common (x – 3)
= (x – 3)(x + 2)
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

34. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6 put the four terms into two pairs
= (x2 – 3x) + (2x – 6) take out the GCF of each pair
= x(x – 3) + 2(x – 3)
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

35. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
II. If the table is impossible to do, then the trinomial is prime.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6 put the four terms into two pairs
= (x2 – 3x) + (2x – 6) take out the GCF of each pair
= x(x – 3) + 2(x – 3) take out the common (x – 3)
= (x – 3)(x + 2)
Here is an example of factoring a trinomial by grouping.
Factoring Trinomials and Making Lists

36. The ac-Method (for factoring trinomial)
A table like the ones above can be made from a given trinomial
and the ac–method uses the table to check if the given
trinomial is factorable or prime.
I. If we find the u and v that fit the table then it is factorable,
and we may use the grouping method, with the found u and v,
to factor the trinomial.
II. If the table is impossible to do, then the trinomial is prime.
Example B. Factor x2 – x – 6 by grouping.
x2 – x – 6 write –x as –3x + 2x
= x2 – 3x + 2x – 6 put the four terms into two pairs
= (x2 – 3x) + (2x – 6) take out the GCF of each pair
= x(x – 3) + 2(x – 3) take out the common (x – 3)
= (x – 3)(x + 2)
Here is an example of factoring a trinomial by grouping.
Let’s see how the X–table is made from a trinomial.
Factoring Trinomials and Making Lists

37. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
ac
b
Factoring Trinomials and Making Lists

38. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
uv = ac
u + v = b
ac
b
u v
Factoring Trinomials and Making Lists

39. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
ac
b
u v
Factoring Trinomials and Making Lists

40. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
–6
–1
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
In example B, the ac-table for 1x2 – x – 6 is:
ac
b
u v
Factoring Trinomials and Making Lists

41. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
–6
–1
–3 2
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
In example B, the ac-table for 1x2 – x – 6 is:
We found –3, 2 fit the table, so we write
x2 – x – 6 as x2 – 3x + 2x – 6
ac
b
u v
Factoring Trinomials and Making Lists

42. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
–6
–1
–3 2
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
then factor (ax2 + ux) + (vx + c) by the grouping method.
In example B, the ac-table for 1x2 – x – 6 is:
We found –3, 2 fit the table, so we write
x2 – x – 6 as x2 – 3x + 2x – 6
ac
b
u v
Factoring Trinomials and Making Lists

43. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
–6
–1
–3 2
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
then factor (ax2 + ux) + (vx + c) by the grouping method.
In example B, the ac-table for 1x2 – x – 6 is:
We found –3, 2 fit the table, so we write
x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping
= (x2 – 3x) + (2x – 6)
ac
b
u v
Factoring Trinomials and Making Lists

44. ac-Method: Given the trinomial ax2 + bx + c,
it’s ac–table is:
ac at the top,
with b at the bottom,
and we are to find u and v such that
–6
–1
–3 2
uv = ac
u + v = b
I. If u and v are found (so u + v = b),
write ax2 + bx + c as ax2 + ux + vx + c,
then factor (ax2 + ux) + (vx + c) by the grouping method.
In example B, the ac-table for 1x2 – x – 6 is:
We found –3, 2 fit the table, so we write
x2 – x – 6 as x2 – 3x + 2x – 6 and by grouping
= (x2 – 3x) + (2x – 6)
= x(x – 3) + 2(x – 3)
= (x – 3)(x + 2)
ac
b
u v
Factoring Trinomials and Making Lists

45. Example C. Factor 3x2 – 4x – 20 using the ac-method.
Factoring Trinomials and Making Lists

46. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
–60
–4
Factoring Trinomials and Making Lists

47. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
–60
–4
Factoring Trinomials and Making Lists

48. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists

49. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20

50. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups

51. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups
= 3x(x + 2) – 10 (x + 2) pull out common factor

52. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups
= 3x(x + 2) – 10 (x + 2) pull out common factor
= (3x – 10)(x + 2) pull out common factor

53. Example C. Factor 3x2 – 4x – 20 using the ac-method.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups
= 3x(x + 2) – 10 (x + 2) pull out common factor
= (3x – 10)(x + 2) pull out common factor
Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)

54. Example C. Factor 3x2 – 4x – 20 using the ac-method.
If the trinomial is prime then we have to justify it’s prime.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups
= 3x(x + 2) – 10 (x + 2) pull out common factor
= (3x – 10)(x + 2) pull out common factor
Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)

55. Example C. Factor 3x2 – 4x – 20 using the ac-method.
If the trinomial is prime then we have to justify it’s prime.
We do this by listing all the possible u’s and v’s with uv = ac,
and showing that none of them fits the condition u + v = b.
We have that a = 3, c = –20 so ac = 3(–20) = –60,
b = –4 and the ac–table is:
We need two numbers u and v such that
uv = –60 and u + v = –4.
By trial and error we see that 6 and –10 is the
solution so we may factor the trinomial by grouping.
–60
–4
–10 6
Factoring Trinomials and Making Lists
Using 6 and –10, write 3x2 – 4x – 20 as 3x2 + 6x –10x – 20
= (3x2 + 6x ) + (–10x – 20) put in two groups
= 3x(x + 2) – 10 (x + 2) pull out common factor
= (3x – 10)(x + 2) pull out common factor
Hence 3x2 – 4x – 20 = (3x – 10)(x + 2)

56. 1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1
4. 3x2 + 2x – 1 5. 2x2 – 3x + 1 6. 2x2 + 3x – 1
8. 2x2 – 3x – 27. 2x2 + 3x – 2
15. 6x2 + 5x – 6
10. 5x2 + 9x – 2
9. 5x2 – 3x – 2
12. 3x2 – 5x – 211. 3x2 + 5x + 2
14. 6x2 – 5x – 613. 3x2 – 5x – 2
16. 6x2 – x – 2 17. 6x2 – 13x + 2 18. 6x2 – 13x + 2
19. 6x2 + 7x + 2 20. 6x2 – 7x + 2 21. 6x2 – 13x + 6
22. 6x2 + 13x + 6 23. 6x2 – 5x – 4 24. 6x2 – 13x + 8
25. 6x2 – 13x – 8 25. 4x2 – 9 26. 4x2 – 49
27. 25x2 – 4 28. 4x2 + 9 29. 25x2 + 9
Exercise. Factor the following trinomials if possible.
Factoring Trinomials and Making Lists

57. 1. (3x + 2)(x – 1) 3. (3x + 1)(x – 1)
7. (2x – 1)(x + 2) 9. (5x + 2)(x – 1) 11. (3x + 2)(x + 1)
15. (3x – 2)(2x + 3)13. (3x + 1)(x – 2)
15. Non factorable
19. (2x + 1)(3x + 2)
17. (x – 2)(6x – 1)
23. (2x + 1)(3x – 4)21. (2x – 3)(3x – 2)
27. (5x – 2)(5x + 2)25. (2x – 3)(2x + 3)
(Answers to odd problems)
Factoring Trinomials and Making Lists
5. (2x – 1)(x – 1)