Factoring and Operations With Polynomials
Example 1
Perform each operation.
-
(
3
y
2
−
4
y
+
5
)
+
(
y
2
+
9
y
)
open , 3 y squared , minus 4 y plus 5 close plus open , y squared , plus 9 y close
=
(
3
y
2
+
y
2
)
+
(
−
4
y
+
9
y
)
+
5
To add, group like terms.
=
4
y
2
+
5
y
+
5
table with 2 rows and 3 columns , row1 column 1 , equals , column 2 open , 3 y squared , plus , y squared , close plus open negative 4 y plus 9 y close plus 5 , column 3 cap to addcomma group like terms. , row2 column 1 , equals , column 2 4 y squared , plus 5 y plus 5 , end table
-
=
n
(
n
)
+
n
(
−
3
)
+
4
(
n
)
+
4
(
−
3
)
Distribute
n
and
4
.
=
n
2
−
3
n
+
4
n
−
12
Combine like terms.
=
n
2
+
n
−
12
table with 3 rows and 3 columns , row1 column 1 , equals , column 2 n open n close plus n open negative 3 close plus 4 open n close plus 4 open negative 3 close , column 3 cap distribute . n , and , 4 . , row2 column 1 , equals , column 2 n squared , minus 3 n plus 4 n minus 12 , column 3 cap combine like terms. , row3 column 1 , equals , column 2 n squared , plus n minus 12 , end table
To factor a polynomial, first find the greatest common factor (GCF) of the terms. Then use the distributive property to factor out the GCF.
Example 2
Factor
6
x
3
−
12
x
2
+
18
x
.
6 x cubed , minus , 12 x squared , plus 18 x .
6
x
3
=
6
·
x
·
x
·
x
;
−
12
x
2
=
6
·
(
−
2
)
·
x
·
x
;
18
x
=
6
·
3
·
x
List the factors of each term. The GCF is
6
x
.
6
x
3
−
12
x
2
+
18
x
=
6
x
(
x
2
)
+
6
x
(
−
2
x
)
+
6
x
(
3
)
Use the distributive property to factor out
6
x
.
=
6
x
(
x
2
−
2
x
+
3
)
table with 3 rows and 4 columns , row1 column 1 , 6 x cubed , equals 6 middle dot x middle dot x middle dot x semicolon negative , 12 x squared , column 2 equals , column 3 6 middle dot open negative 2 close middle dot x middle dot x semicolon 18 x equals 6 middle dot 3 middle dot x , column 4 cap list the factors of each term. cap the cap gcap ccap f is . 6 x . , row2 column 1 , 6 x cubed , minus , 12 x squared , plus 18 x , column 2 equals , column 3 6 x open , x squared , close plus 6 x open negative 2 x close plus 6 x open 3 close , column 4 cap use the distributive property to factor out . 6 x . , row3 column 1 , , column 2 equals , column 3 6 x open , x squared , minus 2 x plus 3 close , end table
When a polynomial is the product of two binomials, you can work backward to find the factors.
Example 3
Factor
x
2
−
13
x
+
36
.
x squared , minus 13 x plus 36 .
Choose numbers that are factors of 36. Look for a pair with the sum
−
13
.
negative 13 .
The numbers
−
4
negative 4 and
−
9
negative 9 have a product of 36 and a sum of
−
13
.
negative 13 . The factors are
(
x
−
4
)
open x minus 4 close and
(
x
−
9
)
.
open x minus 9 close . So,
x
2
−
13
x
+
36
=
(
x
−
4
)
(
x
−
9
)
.
x squared , minus 13 x plus 36 equals open x minus 4 close open x minus 9 close .
| Factors |
Sum |
|
−
6
·
(
−
6
)
negative 6 middle dot open negative 6 close
|
−
12
negative 12
|
|
−
4
·
(
−
9
)
negative 4 middle dot open negative 9 close
|
−
13
negative 13
|
Exercises
Perform the indicated operations.
-
(
x
2
+
3
x
−
1
)
+
(
7
x
−
4
)
open , x squared , plus 3 x minus 1 close plus open 7 x minus 4 close
-
(
5
y
2
+
7
y
)
−
(
3
y
2
+
9
y
−
8
)
open , 5 y squared , plus 7 y close minus open , 3 y squared , plus 9 y minus 8 close
-
4
x
2
(
3
x
2
−
5
x
+
9
)
4 x squared , open , 3 x squared , minus 5 x plus 9 close
-
−
5
d
(
13
d
2
+
7
d
+
8
)
negative 5 d open , 13 d squared , plus 7 d plus 8 close
-
(
x
−
5
)
(
x
+
3
)
open x minus 5 close open x plus 3 close
-
(
n
−
7
)
(
n
−
2
)
open n minus 7 close open n minus 2 close
Factor each polynomial.
-
a
2
−
8
a
+
12
eh squared , minus 8 eh plus 12
-
n
2
−
2
n
−
8
n squared , minus 2 n minus 8
-
x
2
+
5
x
+
4
x squared , plus 5 x plus 4
-
3
m
2
−
9
3 m squared , minus 9
-
y
2
+
5
y
−
24
y squared , plus 5 y minus 24
-
s
3
+
6
s
2
+
11
s
s cubed , plus , 6 s squared , plus 11 s
-
2
x
3
+
4
x
2
−
8
x
2 x cubed , plus , 4 x squared , minus 8 x
-
y
2
−
10
y
+
25
y squared , minus 10 y plus 25
