There are three boxes containing fruits. Items can be taken from each box, but the same items must be taken from each box. The greatest common factor (GCF) is defined as the largest quantity that can be taken from each box. The document then provides examples of finding the GCF of various mathematical expressions, such as finding the GCF of 24 and 36 to be 12, and the GCF of x3y5, x4y6, and x5y4 to be x3y4.

1. Factoring Out GCF

2. There are three boxes of fruits as shown here.
Factoring Out GCF

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

4. There are three boxes of fruits as shown here.
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,
Factoring Out GCF

5. There are three boxes of fruits as shown here.
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,
Factoring Out GCF

6. There are three boxes of fruits as shown here.
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 three bananas and two carrot.
Factoring Out GCF

7. There are three boxes of fruits as shown here.
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 three bananas and two carrot.
A group of items which may be taken from each of the three
boxes is a group of common items.
Factoring Out GCF

8. There are three boxes of fruits as shown here.
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 three 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
Factoring Out GCF

9. There are three boxes of fruits as shown here.
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 three 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.
Factoring Out GCF

10. To factor x means to rewrite the quantity x = a*b as a product.
Factoring Out GCF

11. To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime.
Factoring Out GCF

12. To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime.
Factoring Out GCF
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

13. To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

14. Example A. Factor 12 completely.
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
The numbers
2, 3, 5, 7,. . 11, . .
are prime.

15. Example A. Factor 12 completely.
12 = 3 * 4
not prime
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
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 = 3 * 2 * 2
not prime factored completely
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
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 x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
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.

18. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, so 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.

19. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, so 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.

20. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, so 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.

21. Example A. Factor 12 completely.
12 = 3 * 4 = 3 * 2 * 2
not prime factored completely
To factor x means to rewrite the quantity x = a*b as a product.
A quantity x that can’t be written as a product, besides as 1*x,
is said to be prime. To factor completely means to factor x so
each factor in the product is prime.
Factoring Out GCF
Example B.
a. Since 6 = 2*3, 15 = 3*5, so 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.

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

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
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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} =
b. GCF{4ab, 6a} =
c. GCF {6xy2, 15 x2y2} =
d. GCF{x3y5, x4y6, x5y4} =
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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} =
c. GCF {6xy2, 15 x2y2} =
d. GCF{x3y5, x4y6, x5y4} =
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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} =
d. GCF{x3y5, x4y6, x5y4} =
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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.
d. GCF{x3y5, x4y6, x5y4} =
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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} = x3y4.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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.
The Extraction Law
Distributive law interpreted backward gives the Extraction Law,
that is, common factors may be extracted from sums or
differences.
AB ± AC
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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.
AB ± AC  A(B±C)
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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)
This procedure is also called “factoring out a common factor”.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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 any expression, the first step is always to factor out
the GCF, then factor the “left over” as needed.
The GCF of a list of numbers
is the largest number that
may be divided by all the number.

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

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

35. 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 6x2y2)
Example D. Factor out the GCF.
a. xy – 4y = y(x – 4) or (x - 4)y
b. 4ab + 6a
Note the order
of the “( )’s”
doesn’t matter
because AB=BA.

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

37. Factoring Out GCF
(the GCF is y)
(the GCF is 2a)
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 =

38. Factoring Out GCF
(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)

39. Factoring Out GCF
(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)

40. 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)

41. 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) or (x - 4)y
b. 4ab + 6a = 2a(2b) + 2a(3) = 2a(2b + 3)
c. 12x2y3 + 6x2y2 = 6x2y2(2y) + 6x2y2(1) = 6x2y2(2y + 1)

42. 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)
(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)

43. 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)

44. 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)

45. 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.

46. 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 the
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

47. 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 the
common parenthesis.
Example F. Factor by pulling out twice.
a. 3x – 3y + ax – ay
Factor by Grouping

48. 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 the
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

49. 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 the
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

50. 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 the
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

51. We may need to pull out the negative sign
e.g. writing –4x + 10 as –2(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 the
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

52. We may need to pull out the negative sign
e.g. writing –4x + 10 as –2(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 the
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
write –4x + 10 as –2(2x – 5),

53. We may need to pull out the negative sign
e.g. writing –4x + 10 as –2(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 the
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
write –4x + 10 as –2(2x – 5),

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

55. 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

56. 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

57. 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 trinomial x2 – 3x + 2 by grouping by writing
x2 – 3x + 2 = x2 – 2x – x + 2
Factor by Grouping

58. 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 trinomial x2 – 3x + 2 by grouping by writing
x2 – 3x + 2 = x2 – 2x – x + 2
x2 – 3x + 2 = x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2)
Factor by Grouping

59. 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 trinomial x2 – 3x + 2 by grouping by writing
x2 – 3x + 2 = x2 – 2x – x + 2
x2 – 3x + 2 = x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2)
= x(x – 2) –1(x – 2)
Factor by Grouping

60. 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 trinomial x2 – 3x + 2 by grouping by writing
x2 – 3x + 2 = x2 – 2x – x + 2
x2 – 3x + 2 = x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2)
= x(x – 2) –1(x – 2) Pull the factor (x – 2) again.
= (x – 2 )(x – 1)
Factor by Grouping

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
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 trinomial x2 – 3x + 2 by grouping by writing
x2 – 3x + 2 = x2 – 2x – x + 2
x2 – 3x + 2 = x2 – 2x – x + 2
= (x2 – 2x) + (–x + 2)
= x(x – 2) –1(x – 2) Pull the factor (x – 2) again.
= (x – 2 )(x – 1)
In next section we use the grouping method to factor a given
trinomial or determine that it’s prime.
Factor by Grouping

62. 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

63. 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