2.01
Factoring, Factoring, Factoring
from
Intermediate
Algebra: One Step at a Time
© 1998
p. 111142
Dr.
Robert J. Rapalje
Seminole Community College
PLEASE NOTE: Links are now
provided throughout this section for additional explanations!!
Of all the topics that you have studied in previous algebra courses, there
is no topic more important, there is no topic that you need to review more
than the topic of factoring. Let's begin with a working definition
of factoring. What does it mean to factor something? What would
you say if you were asked to "factor the number 15"? Without
hesitation, you would probably answer "3 times 5" or "5 times
3"! The key word is "times." When asked to factor a
given number, you naturally answer with a product of two numbers.
DEFINITION
To FACTOR means to EXPRESS AS A PRODUCT!
Factoring is an important skill that goes all the way back to your first
algebra course, and it will continue to be a most important skill,
especially in calculus. There are many different types of factoring that
become more complex as well as more abstract in the higher math. Some of
the exercises in this section will begin with a review of elementary
factoring exercises, such as
x2
9 or
x2
6
x,
and then progress into higher levels of factoring. Notice the increase
in complexity and abstraction as you "grow" through this "one step at a
time."
GUIDELINES
TO FACTORING
1. Common Factor
(Factor Common Factor First!)
2. Trinomial (F OI L rearranged to spell F L OI)
3.
Difference of Squares: X2 Y2 = (X Y)(X + Y)
Diff of Cubes: X3 Y3 = (X Y)(X2
+ XY + Y2)
Sum
of Cubes: X3 + Y3 = (X + Y)(X2
XY + Y2)
4. Factoring by Grouping
FACTORING
THE COMMON FACTOR
See also :
Basic Algebra: Factoring the Common Factor
Intermediate Algebra: Factoring the Common Factor in Living Color
The first step in any factoring problem should be to try to factor
the common factor. Factoring the common factor is simply using the
distributive property in reverse.
EXAMPLES
DISTRIBUTIVE PROPERTY:
EXAMPLES OF FACTORING
6(x
+ 7) = 6
x
+ 42
6
x
+ 42 = 6(x
+ 7)
7(2
x
+ 3) = 14
x
+ 21
14
x
+ 21 = 7(2
x
+ 3)
9(3
x
4y) = 27
x
36y
27
x
36y
= 9(3
x
4y)
12(2
x
+ 1) = 24
x
+ 12
24
x
+ 12
= 12(2
x
+ 1)
5(3
x
2y + 4) = 15
x
10y + 20
15
x
10Y + 20
= 5(3
x
2Y + 4)
5
x
(x
+ 4) = 5
x2
+ 20
x
5
x2
+ 20
x
= 5
x
(x
+ 4)
When factoring the common factor, look for a number or variable that
divides into both (or all) terms. If there is more than one common
factor, be sure to get the largest common factor you can find. First
write down the common factor. Then, open parentheses, and put down all
the other factors that are left over. Again, follow the one step format
of the following exercises.
EXERCISES: Factor completely.
1. 5
x2
+ 15
x
2. 18
x
+ 24y
3. 35
x y
+ 7
y
= 5
x
( )
= 6(
)
= 7
y
(
)
4. 5
x3
45
x2
5. 16a + 24b 8
6. 12
x
24
y
+ 48
= 5
x
2(
) = 8(
)
= 12(
)
7.
x3
+ 4
x2
8. 4
y
3
+ 8
y
9. 16a3
24a2
10. 12z3
18z2
11. 24
x2
+ 12
12. 16b2
+ 48b3
13. 24
x3
+ 24
x2
14. 16
x2
32
x3
15. 8a
+ 12b 20c
16. 40
x
32
y
+ 64
17. 42
y3
14
y2
+ 49
y
18. 24
x3
+ 24
x2
+ 24
x
19. 19
x3
+ 19
x2
y
+ 38
x2
20. 17
x3
34
x2
21.
y
5
14y3
22.
x10
+ 5
x3
=
y
3(
)
= x3(
)
From these examples, observe the rule listed below:
RULE
When factoring powers, take out the lowest power of the
factor. Then subtract exponents.
|
23.
y10
+ 7
y4
24.
x7
+ 8
x5
25. 16
x2
y3
12
x3
y2
= 4
x2
y2(
)
26. 5
x5
y2
+ 10
x4
y3
27.
8
x5
y3
+ 12
x3
y4
28. 36
x3
y4
+ 24
x2
y6
In each of the next exercises, observe how you move from the simple to the
more complicated; from the concrete to the abstract.
29a)
y x
+ 7
x
30a) 4
x y
+ 3
y
=
x(
)
b)
ya
+ 7a
b) 4
xa
+ 3a
= a(
)
c) y$ + 7$
c) 4
x$
+ 3$
= $(
)
d) y(Junk) + 7(Junk)
d) 4X(Junk) + 3(Junk)
= (Junk) (
)
e) y(3
x+4)
+ 7(3
x+4)
e) 4
x(8
y7)
+ 3(8
y7)
= (3
x+4)
(
)
31. a(3
x+4)
5(3
x+4)
32. 5q(8r+7) + 3(8r+7)
33. 5u(3
x+4)
+ 9v(3
x+4)
34. 10x(8
y7)
3(8
y7)
RULE
In order to factor a common factor, you must have an identical
factor common to all terms. Be sure to count terms first.
|
EXAMPLE:
Can you factor 5u(3
x+4)
+ 9v(3
x4)
in this manner?
NO! There is no factor common to both terms.
35.
x(x
y)
y
(x
y)
36.
x(x
y)
y(x
y)
+ 4(x
y)
37.
x(x
y)
+ y(x
y)
4(x
y)
38.
x(2
x+3
y)
y(2
x+3
y)
+ 4(2
x+3
y)
39. (x
+
y)2
z(x
+
y)
40. (x
y)2
z(x
y)
= ( )[(
) ___]
= ( )(
)
41. (x
y)2
y(x
y)
42. (x
+
y)2
y(x
+
y)
43. (2
x
+ 3
y)2
5(2
x
+ 3
y)
44. (2
x
3
y)2
5(2
x
3
y)
45. (2
x
+ 3
y)2
+ 5(2
x
+ 3
y)
46. (x
+
y)2
+ (x
y)(
x
+
y)
47. (3
x2
y)2
2(3
x2
y)(
x5
y)
48. (5
x+3
y)2
4(5
x+3
y)(
x+3
y)
TRINOMIALS
See also :
Basic Algebra: Factoring Trinomials
Do you remember the product of two binomials, F OI L, and the fact
that the result is usually in a trinomial? As examples, consider:
Product Binomials F
OI L
(x
+ 2)(
x
+ 5) =
x2
+ 7x
+ 10
(x
+ 2)(
x
5) =
x2
3x
10
(x
2)(
x
+ 5) =
x2
+ 3x
10
(x
2)(
x
5) =
x2
7x
+
10
In each of these examples, you were given a product of two
binomials, and with F
OI
L, in each case you
obtained a trinomial with
x2.
Now the problem will be to work these problems in reverse. What if you
were given a trinomial, such as
x2+7
x+10
and asked to factor itthat is, to express it as a product.
This is the product of the two binomials. When you factor
the trinomial:
x2
+ 7
x
+ 10
you expect the product of binomials: (
)( ).
When factoring a trinomial, instead of thinking
F
OI L,
you need to change the order and think F
L
OI.
In other words, you need to find the correct
F
(first times first) combination, then skip to the
L
(last times last). Finally, check to make sure the
OI (outer
times outer, inner times inner) middle term is correct.
F
OI
L
x2
+
7
x
+ 10 Given
trinomial to factor;
(
)( )
Product of two binomials;
(x
)(
x
)
F term is
x2,
which is
x
times
x;
(x
)(x
)
L term is +10.
Find two numbers whose product is +10. Probably 2 times 5, or it
could be 1 times 10. Try 2 times 5. Since the la
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