Lecture 3–3A Factoring Out Common Factors with Negative Exponents
When you factor out a common factor from a polynomial the answer contains a distributive expression x
2 + x
= x ( x
+
1) 6 x
5 −
9 x
4 =
3 x
4
(2 x
−
3)
The common factor is placed outside a bracket and what remains inside the bracket is found by dividing each term by the common factor
Example 1 Example 2
Factor a x out of x
2 + x Factor a 3x
4
out of 6 x
5 −
9 x
4
= x
⎛
⎝ x x
2
+
= x ( x
+
1) x x
⎞
⎠
=
3 x
4
⎛
⎝
6 x
3 x
4
5
−
9 x
4
3 x
4
⎞
⎠
=
3 x
4
(2 x
−
3)
If you have several terms with x to a power the lowest power of x is the common factor. The lowest power of x will be divided out and the remaining polynomial inside the parenthesis will be simplified using the Laws of Exponents.
In the problems below the lowest power of x is a negative number.
Factor out the lowest power of x and simplify the remaining polynomial inside the parenthesis using the Laws of
Exponents. If the coefficient also has a common factor, be sure and factor that out as well.
Example 3
Factor a x
−
2
out of x
3 + x
−
2
= x
−
2
⎛
⎝ x x
3
−
2
+ x x
−
2
−
2
⎞
⎠
= x
−
2
( x
3 • x
2 + x
−
2 • x
2
)
= x
−
2
( x
5 + x
0
)
= x
−
2 5 +
1
= x
5 +
1 x
2
Math 400 3–3A Factoring Out Least Powers Page 1 of 4
© 2013
Eitel

Example 4 factor out a x
−
1 2
3 from x 3 + x
−
1
3
= x
−
1
3
⎛
⎜
⎜
⎝ x
2 x 3
−
1
3
+ x x
−
1
3
−
1
3
⎞
⎟
⎟
⎠
= x
−
1
3
⎛
⎜ x
2
3
1
• x 3 + x
−
1
3 • x
1
3
⎞
⎟
= x
−
1
3
⎛
⎜ x
3
3 + x
0
⎞
⎟
= x
−
1
3
( x
+
1
)
=
( x
+
1
)
1 x 3
=
( x
+
1
)
3 x
Math 400 3–3A Factoring Out Least Powers Page 2 of 4
© 2013
Eitel

x
−
1 x
−
2
= x
Dividing out a common x term using the power rule requires us to subtract exponents.
Subtracting a negative exponent creates an addition problem
Quotient Rule
−
1
+
2 = x
Quotient Rule x a x b
= x a
− b
Quotient Rule x x
2
5
−
3
5
= x
2
5
+
3
5 = x
Quotient Rule
( x
−
5)
( x
−
5)
−
3
−
4
=
( x
−
5)
−
3
+
4 =
( x
−
5)
This technique may be easier for many problems.
Example 5 factor out a x
−
3
5 from 2 x
2
5
−
7 x
−
3
5
= x
−
3
5 ⎜
⎝
⎛
⎜
2
• x
2
5
+
3
5 −
7
• x
−
3
5
+
3
5
⎞
⎟
= x
−
3
5
(
2 x
−
7
)
=
(
2 x
−
7
)
3 x 5
=
(
2 x
−
7
)
5 x
3
Math 400 3–3A Factoring Out Least Powers Page 3 of 4
© 2013
Eitel

Example 6 factor out a x
−
1
3 from 4 x
2
3
−
9 x
−
1
3
= x
−
1
3 ⎜
⎝
⎛
⎜
4
• x
2
3
+
1
3 −
9
• x
−
1
3
+
1
3
⎞
⎟
⎟
⎠
= x
−
1
3
(
4 x
−
9
)
=
(
4 x
−
9
)
1 x 3
=
(
2 x
−
7
)
3 x
Example 7 factor out a ( x
−
1)
−
3
from 2( x
−
1)
−
2 −
5( x
−
1)
−
3
=
( x
−
1)
−
3
⎛
⎝
2( x
−
1)
( x
−
1)
−
2
−
3
−
5( x
−
1)
( x
−
1)
−
3
−
3
⎞
⎠
=
( x
−
1)
−
3 (
2( x
−
1)
−
5
)
=
( x
−
1)
−
3 (
2 x
−
2
−
5
)
=
2 x
−
7
( x
−
1)
3
Math 400 3–3A Factoring Out Least Powers Page 4 of 4
© 2013
Eitel