LESSON 7.7
ADDING
AND
SUBTRACTING
POLYNOMIALS
California
Standards
10.0 Students add, subtract, multiply, and
divide monomials and polynomials. Student
solve multistep problems, including word
problems, by using these techniques.
Remember!
Like terms are constants or terms with the
same variable(s) raised to the same
power(s).
Adding Polynomials
Horizontal Method
(4x  x  5x  7)  (8x  2x  1)
3
2
3
4x  x  7x  6
3
2
Vertical Method
4x  x  5x  7
3
8x
 2x  1
3
2
4x  x  7x  6
3
2
Add the polynomials.
1) (6x  4x  3)  (9x  6x 11)
3
3
15x  2x  8
3
2) (7y  9y  6)  (3y 10y  8)
4
2
3
2
7y  3y  y  14
4
3
2
3) 4x 2  5x  2
2
3x  4x  7
4) 6m 3  2m 2  m
3
m
 7m  10
7x  9x  5
5m 2m  6m  10
2
3
2
Remember!
When you use the Associative and
Commutative Properties to rearrange
the terms, the sign in front of each
term must stay with that term.
Opposite of a Polynomial
(a  b  c)  a  b  c
Simplify.
1)  (x  4x  9)
3
3)  (t  t  9t  2)
4
3
 x  4x  9
 t  t  9t  2
2)  (y  2y  7y)
4)  (5y  8y  9y  3)
3
5
3
 y  2y  7y
5
3
3
4
3
2
 5y3  8y2 9y  3
NOW YOU TRY…
1) (y  5y  6)  (y  2y  9)
2
2
y  5y  6  y  2y  9
2
2
3y  15
2) (6m  4m  1)  (m  2m  6)
2
2
6m  4m  1  m  2m  6
2
2
5m  6m  7
2
Simplify.
3) (3p  4p  5)  (2p  p  7p)
3
2
3
2
3p  4p  5  2p  p  7p
3
3
2
2
p  5p  7p  5
3
2
4) (9t  6t  8)  (4t  t)
2
2
9t  6t  8  4t  t
2
2
13t  5t  8
2
Simplify.
5) (7x  3x  4)  (9x  4x  3)
2
2
16x 2  x  1
6) (6t  8t  t  t)  (t  6t  5t  8)
4
3
2
3
2
6t 4  8t 3  t 2  t  t 3  6t 2  5t  8
6t  9t  5t  6t  8
4
3
2
Application
A farmer must add the areas of two plots of land
to determine the amount of seed to plant. The
area of plot A can be represented by 3x2 + 7x –
5, and the area of plot B can be represented by
5x2 – 4x + 11. Write a polynomial that
represents the total area of both plots of land.
(3x2 + 7x – 5)
+ (5x2 – 4x + 11)
8x2 + 3x + 6
Plot A.
Plot B.
Combine like terms.
Write an expression that represents the area of
the shaded region in terms of x.
x+2
1)
2)
3
9
6
55
xx ++ 22
2x + 5
3x + 7
6(2x  5)  3(x  2)
9(3x  7)  5(x  2)
12x  30  3x  6
27x  63  5x  10
15x  36
22x  53
Write an expression that represents the area of
the shaded region in terms of x.
2
2x
4
2
3x

2x
1)
2)
5
xx2288
8
7
33
6x 2  5x
7(6x  5x)  5(3x  2x)
8(2x  4)  3(x 2  8)
42x  35x  15x  10x
16x  32  3x  24
2
2
2
2
57x  25x
2
2
2
2
13x  56
2
Lesson Quiz: Part I
Add or subtract.
1. 7m2 + 3m + 4m2
11m2 + 3m
2. (r2 + s2) – (5r2 + 4s2)
–4r2 – 3s2
3. (10pq + 3p) + (2pq – 5p + 6pq) 18pq – 2p
4. (14d2 – 8) + (6d2 – 2d + 1) 20d2 – 2d – 7
5. (2.5ab + 14b) – (–1.5ab + 4b)
4ab + 10b
Lesson Quiz: Part II
6. A painter must add the areas of two walls to
determine the amount of paint needed. The area
of the first wall is modeled by 4x2 + 12x + 9, and
the area of the second wall is modeled by
36x2 – 12x + 1. Write a polynomial that
represents the total area of the two walls.
40x2 + 10