Notes for Lesson 7-6: Adding and Subtracting Polynomials
7-6.1 – Adding and Subtracting Monomials
Earlier in the year, we added and subtracted like terms. We said like terms were terms that had the same variables at the same power. These term were also monomials. So to add monomials we simply add like terms.
Examples: Add or subtract.
15 m
3 
6 m
2 
2 m
3
17 m
3 
6 m
2
3
 x
4
2 x

2
5


7
17 x
2 
12 0 .
9 y
5
1 .
5 y
5

0 .
4 y
5 
0 .
5 x
5  y
5
 x
5
7-6.2 – Adding Polynomials
To add polynomials we can either line up the polynomials vertically by like terms or we can use the associative and commutative properties to collect like terms and add horizontally.
Examples: Add.
( 2 x
2  x )

( x
2 
3 x

1 )
(

2 ab
 b )

( 2 ab
 a )
( 4 x
2 
3 x

6 )

( 2 x
2

2 x
2  x x
2 
3 x

1

2 ab

2 ab
 b
 a

4 x
2 
3 x

6
2 x
2 
4 x

5

4 x

5 )
3 x
2 
2 x

1 0 b
 a 6 x
2  x

1 or
( 2 x
2  x
2
)

(
 x

3 x )

(

1 ) or
(

2 ab

2 ab )

( b )

( a ) b
 a or
( 4 x
2 
2 x
2
)

( 3 x

4 x )

(

6

5 )
3 x
2 
2 x

1
7-6.3 – Subtracting Polynomials
6 x
2  x

1
To subtract polynomials you must remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial you must write the opposite of each term in the polynomial. Then add the polynomials together using the above method.
Examples: Subtract.
( a 4 
2 a )

( 3 a 4 
3 a

1 ) ( 3 x
2 
2 x

8 )

( x
2 
4 ) ( 2 x
2 
3 x
2 
1 )

( x
2  x

1 )
( a
4 
2 a )

(

3 a
4 
3 a

1 ) ( 3 x
2 
2 x

8 )

(
 x
2 
4 ) (
 x
2 
1 )

(
 x
2  x

1 ) a
4 
2 a 3 x
2 
2 x

8
 x
2 
1
 
3 a
4 
3 a

1

2 a 4  a

1
  x
2 
4
2 x
2 
2 x

12
  x
2  x

1

2 x 2  x or
( a
4 
3 a
4
)

(

2 a

3 a )

(

1 )

2 a
4  a

1 or
( 3 x
2  x
2
)

(

2 x )

( 8

4 )
2 x
2 
2 x

12 or
(
 x
2  x
2
)

(
 x )

( 1

1 )

2 x
2  x

7-6.4 – Business Application
The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant.

0 .
Eastern Plant
03 x
2 
25 x

1500

0 .
Western Plant
02 x
2 
21 x

1700
Write a polynomial that represents the difference of the profits at the Eastern plant and the Western plant.
(

0 .
03 x
2 
25 x

1500 )

(

0 .
02 x
2 
21 x

1700 )
(

0 .
03 x
2 
25 x

1500 )

( 0 .
02 x
2 
21 x

1700 )


0 .
03 x
2 
25 x

1500
0 .
02 x
2 
21 x

1700

0 .
01 x
2 
4 x

200
Do Practice B #’s 7, 12, 13, 14