SECT
ION
5.3
5.3
5.3
OBJECTIVES
1. Add two polynomials
2. Subtract two polynomials
Polynomials: Addition
and Subtraction
Addition is always a matter of combining like quantities (two apples plus three apples,
four books plus five books, and so on). If you keep that basic idea in mind, adding
polynomials will be easy. It is just a matter of combining like terms. Suppose that you
want to add
5x2 3x 4
and
4x2 5x 6
Parentheses are sometimes used in adding, so for the sum of these polynomials, we
can write
(5x2 3x 4) (4x2 5x 6)
The plus sign between the
parentheses indicates the
addition.
Now what about the parentheses? You can use the following rule.
Removing Signs of Grouping: Case 1
If a plus sign () or nothing at all appears in front of parentheses, just remove the
parentheses. No other changes are necessary.
Now let’s return to the addition.
Just remove the parentheses.
No other changes are
necessary.
(5x2 3x 4) (4x2 5x 6)
5x2 3x 4 4x2 5x 6
Like terms
Like terms
Like terms
Note the use of the
associative and commutative
properties in reordering and
regrouping.
Collect like terms. (Remember: Like terms have the same variables raised to the
same power.)
(5x2 4x2) (3x 5x) (4 6)
Combine like terms for the result:
Here we use the distributive
property. For example,
5x2 4x2
( 5 4 ) x2
9x2
352
9x2 8x 2
As should be clear, much of this work can be done mentally. You can then write
the sum directly by locating like terms and combining.
Section 5.3
Example 1
■
Polynomials: Addition and Subtraction
Combining Like Terms
Add 3x 5 and 2x 3.
Write the sum.
(3x 5) (2x 3)
3x 5 2x 3 5x 2
Like
terms
Like terms
✓ CHECK YOURSELF 1
■
Add 6x2 2x and 4x2 7x.
The same technique is used to find the sum of two trinomials.
Example 2
Adding Polynomials
Add 4a2 7a 5 and 3a2 3a 4.
Write the sum.
(4a2 7a 5) (3a2 3a 4)
4a2 7a 5 3a2 3a 4 7a2 4a 1
Remember: Only the like
terms are combined in the
sum.
Like terms
Like terms
Like terms
✓ CHECK YOURSELF 2
■
Add 5y2 3y 7 and 3y2 5y 7.
Example 3
Adding Polynomials
Add 2x2 7x and 4x 6.
Write the sum.
353
354
Chapter 5
■
Polynomials
(2x2 7x) (4x 6)




2x2 7x 4x 6
These are the only like terms;
2x2 and 6 cannot be combined.
2x2 11x 6
✓ CHECK YOURSELF 3
■
Add 5m2 8 and 8m2 3m.
As we mentioned in Section 5.2, writing polynomials in descending-exponent form
usually makes the work easier. Look at Example 4.
Example 4
Adding Polynomials
Add 3x 2x2 7 and 5 4x2 3x.
Write the polynomials in descending-exponent form, then add.
(2x2 3x 7) (4x2 3x 5)
2x2 12
✓ CHECK YOURSELF 4
■
Add 8 5x2 4x and 7x 8 8x2.
Subtracting polynomials requires another rule for removing signs of grouping.
Removing Signs of Grouping: Case 2
If a minus sign () appears in front of a set of parentheses, the parentheses can
be removed by changing the sign of each item inside the parentheses.
The use of this rule is illustrated in Example 5.
Section 5.3
Example 5
Polynomials: Addition and Subtraction
■
355
Removing Parentheses
In each of the following, remove the parentheses.
Note: This uses the
distributive property, since
Change each sign to remove the parentheses.
(b) m (5n 3p) m 5n 3p





(2x 3y) (1)(2x 3y)
2x 3y
(a) (2x 3y) 2x 3y
Sign changes.





(c) 2x (3y z) 2x 3y z
Sign changes.
✓ CHECK YOURSELF 5
■
Remove the parentheses.
(a) (3m 5n)
(b) (5w 7z)
(c) 3r (2s 5t)
(d) 5a (3b 2c)
Subtracting polynomials is now a matter of using the previous rule to remove the
parentheses and then combining like terms. Consider Example 6.
Example 6
Note: The expression
following “from” is written
first in the problem.
Subtracting Polynomials
(a) Subtract 5x 3 from 8x 2.
Write
(8x 2) (5x 3)





8x 2 5x 3
Sign changes.
3x 5
(b) Subtract 4x2 8x 3 from 8x2 5x 3.
Write
(8x2 5x 3) (4x2 8x 3)







8x2 5x 3 4x2 8x 3
Sign changes.
4x2 13x 6
356
Chapter 5
■
Polynomials
✓ CHECK YOURSELF 6
■
(a) Subtract 7x 3 from 10x 7.
(b) Subtract 5x2 3x 2 from 8x2 3x 6.
Again, writing all polynomials in descending-exponent form will make locating
and combining like terms much easier. Look at Example 7.
Example 7
Subtracting Polynomials
(a) Subtract 4x2 3x3 5x from 8x3 7x 2x2.
Write
(8x3 2x2 7x) (3x3 4x2 5x)









8x3 2x2 7x 3x3 4x2 5x
Sign changes.
11x 2x 12x
3
2
(b) Subtract 8x 5 from 5x 3x2.
Write
(3x2 5x) (8x 5)





3x2 5x 8x 5
Only the like terms can be combined.
3x 13x 5
2
✓ CHECK YOURSELF 7
■
(a) Subtract 7x 3x2 5 from 5 3x 4x2.
(b) Subtract 3a 2 from 5a 4a2.
If you think back to addition and subtraction in arithmetic, you’ll remember that the
work was arranged vertically. That is, the numbers being added or subtracted were placed
under one another so that each column represented the same place value. This meant that
in adding or subtracting columns you were always dealing with “like quantities.”
Section 5.3
■
Polynomials: Addition and Subtraction
357
It is also possible to use a vertical method for adding or subtracting polynomials.
First rewrite the polynomials in descending-exponent form, then arrange them one under another, so that each column contains like terms. Then add or subtract in each
column.
Example 8
Adding Using the Vertical Method
(a) Add 3x 5 and x2 2x 4
Like terms
3x 5 2
x2 2x 4
————–—
x2 5x 1
(b) Add 2x2 5x, 3x2 2, and 6x 3.
Like terms
2x2 5x 2
3x2
2
6x 3
————–—
5x2 x 1
✓ CHECK YOURSELF 8
■
Add 3x2 5, x2 4x, and 6x 7.
The following example illustrates subtraction by the vertical method.
Example 9
Subtracting Using the Vertical Method
(a) Subtract 5x 3 from 8x 7.
Write
()8x 7
()5x 3
————–
8x 7
5x 3
————
3x 4
To subtract, change each
sign of 5x 3 to get
5x 3, then add.
358
Chapter 5
■
Polynomials
(b) Subtract 5x2 3x 4 from 8x2 5x 3.
Write
()8x2 5x 3
()5x2 3x 4
———————–
To subtract, change each
sign of 5x2 3x 4 to get
5x2 3x 4, then add.
8x2 5x 3
5x2 3x 4
——————–
3x2 8x 7
Subtracting using the vertical method takes some practice. Take time to study the
method carefully. You’ll be using it in long division in Section 6.4.
✓ CHECK YOURSELF 9
■
Subtract, using the vertical method.
(a) 4x2 3x from 8x2 2x
(b) 8x2 4x 3 from 9x2 5x 7
✓ CHECK YOURSELF ANSWERS
■
1. 10x2 5x.
2. 8y2 8y.
3. 13m2 3m 8.
4. 3x2 11x.
5. (a) 3m 5n; (b) 5w 7z; (c) 3r 2s 5t; (d) 5a 3b 2c.
6. (a) 3x 10; (b) 3x2 8.
8. 4x2 2x 12.
7. (a) 7x2 10x; (b) 4a2 2a 2.
9. (a) 4x2 5x; (b) x2 9x 10.
E xercises
1. 9a 4
2. 12x 1
3. 13b2 18b
4. 8m2 5m
2
2
5. 2x
6. 4p
2
7. 5x 2x 1
2
8. 9d 14d 2
9. 2b 5b 16
2
10. 3x 5x 3
4
11. 8y 2y
3
13. a 4a
12. 9x 3
2
14. 5m3 8m
15. 2x2 x 3
16. 2b3 5b2 3b
17. 2a 3b
18. 7x 4y
19. 5a 2b 3c
20. 7x 4y 3z
21. 6r 5s
22. 7m 2n
23. 8p 2q
24. 7c 10d
25. x 7
26. 2x 7
27. m2 3m 28. 2a2 5a
29. 2y2
5.3
Add.
1. 6a 5 and 3a 9
2. 9x 3 and 3x 4
3. 8b2 11b and 5b2 7b
4. 2m2 3m and 6m2 8m
5. 3x2 2x and 5x2 2x
6. 3p2 5p and 7p2 5p
7. 2x2 5x 3 and 3x2 7x 4
8. 4d2 8d 7 and 5d2 6d 9
9. 2b2 8 and 5b 8
2
3
■
30. 2n2
31. 2x2 x 1
32. 2x2 6x 7
33. 8a2 12a 7
34. x3 x2 5x
10. 4x 3 and 3x2 9x
11. 8y3 5y2 and 5y2 2y
12. 9x4 2x2 and 2x2 3
13. 2a2 4a3 and 3a3 2a2
14. 9m3 2m and 6m 4m3
15. 4x2 2 7x and 5 8x 6x2
16. 5b3 8b 2b2 and 3b2 7b3 5b
Remove the parentheses in each of the following expressions, and simplify where
possible.
17. (2a 3b)
18. (7x 4y)
19. 5a (2b 3c)
20. 7x (4y 3z)
21. 9r (3r 5s)
22. 10m (3m 2n)
23. 5p (3p 2q)
24. 8d (7c 2d)
Subtract.
25. x 4 from 2x 3
26. x 2 from 3x 5
27. 3m2 2m from 4m2 5m
28. 9a2 5a from 11a2 10a
29. 6y2 5y from 4y2 5y
30. 9n2 4n from 7n2 4n
31. x2 4x 3 from 3x2 5x 2
32. 3x2 2x 4 from 5x2 8x 3
33. 3a 7 from 8a2 9a
34. 3x3 x2 from 4x3 5x
35. 4b2 3b from 5b 2b2
36. 7y 3y2 from 3y2 2y
37. x2 5 8x from 3x2 8x 7
38. 4x 2x2 4x3 from 4x3 x 3x2
35. 6b2 8b
36. 6y2 9y 37. 2x2 12
38. x2 3x
359
360
Chapter 5
■
Polynomials
39. 6b 1
Perform the indicated operations.
40. 6m 3
39. Subtract 3b 2 from the sum of 4b 2 and 5b 3.
41. 10x 9
40. Subtract 5m 7 from the sum of 2m 8 and 9m 2.
42. x2 5
41. Subtract 3x2 2x 1 from the sum of x2 5x 2 and 2x2 7x 8.
43. 2x2 5x 12
42. Subtract 4x2 5x 3 from the sum of x2 3x 7 and 2x2 2x 9.
44. 6a 2
43. Subtract 2x2 3x from the sum of 4x2 5 and 2x 7.
45. 6y2 8y
3
2
44. Subtract 5a2 3a from the sum of 3a 3 and 5a2 5.
46. 2r 3r
45. Subtract the sum of 3y2 3y and 5y2 3y from 2y2 8y.
47. 6w2 2w 2
46. Subtract the sum of 7r3 4r2 and 3r3 4r2 from 2r3 3r2.
2
48. 5x 2x 3
49. 9x2 x
Add using the vertical method.
50. 8x2 4x 10
47. 2w2 7, 3w 5, and 4w2 5w
51. 2a2 5a
48. 3x2 4x 2, 6x 3, and 2x2 8
52. 2r3 6r2
49. 3x2 3x 4, 4x2 3x 3, and 2x2 x 7
53. 3x2 x
50. 5x2 2x 4, x2 2x 3, and 2x2 4x 3
54. x2 4x 4
Subtract using the vertical method.
55. 3x2 3x 9
56. 2x2 6x 3
57. 5x2 3x 9
58. 5x2 2x 4
59. (a) 9x 4; (b) 13;
(c) 13
60. (a) 12x2 4x; (b) 8;
(c) 8
61. (a) 7x2 7x; (b) 14;
(c) 14
62. (a) 15x2 x 2; (b) 12;
(c) 12
51. 3a2 2a from 5a2 3a
52. 6r3 4r2 from 4r3 2r2
53. 5x2 6x 7 from 8x2 5x 7
54. 8x2 4x 2 from 9x2 8x 6
55. 5x2 3x from 8x2 9
56. 7x2 6x from 9x2 3
Perform the indicated operations.
57. [(9x2 3x 5) (3x2 2x 1)] (x2 2x 3)
58. [(5x2 2x 3) (2x2 x 2)] (2x2 3x 5)
In Exercises 59 to 66, f(x) and g(x) are given. Let h(x) f(x) g(x). Find (a) h(x),
(b) f(1) g(1), and (c) use the results of part (a) to find h(1).
59. f(x) 5x 3 and g(x) 4x 7
60. f(x) 7x2 3x and g(x) 5x2 7x
61. f(x) 5x2 3x and g(x) 4x 2x2
62. f(x) 8x2 3x 10 and g(x) 7x2 2x 12
Section 5.3
63. (a) x2 2x 2;
(b) 5; (c) 5
3
2
64. (a) 3x 3x 7x 8;
(b) 15; (c) 15
65. 5x3 5x2 11x 25;
(b) 36; (c) 36
66. (a) 3x3 5x2 19x 14; (b) 13; (c) 13
67. (a) 2x 13; (b) 15;
(c) 15
68. (a) 3x 5; (b) 8;
(c) 8
69. (a) 12x2 x; (b) 11;
(c) 11
70. (a) 7x2 6x; (b) 1;
(c) 1
2
71. (a) 3x 2x 7;
(b) 6; (c) 6
72. (a) 2x3 x2 8x 8;
(b) 1; (c) 1
■
Polynomials: Addition and Subtraction
361
63. f(x) 3x2 5x 7 and g(x) 2x2 3x 5
64. f(x) 2x3 5x2 8 and g(x) 5x3 2x2 7x
65. f(x) 5x2 3x 15 and g(x) 5x3 8x 10
66. f(x) 5x2 12x 5 and g(x) 3x3 7x 9
In Exercises 67 to 74, f(x) and g(x) are given. Let h(x) f(x) g(x). Find (a) h(x),
(b) f(1) g(1), and (c) use the results of part (a) to find h(1).
67. f(x) 7x 10 and g(x) 5x 3
68. f(x) 5x 12 and g(x) 8x 7
69. f(x) 7x2 3x and g(x) 5x2 2x
70. f(x) 10x2 3x and g(x) 3x2 3x
71. f(x) 8x2 5x 7 and g(x) 5x2 3x
72. f(x) 5x3 2x2 8x and g(x) 7x3 3x2 8
73. f(x) 5x2 5 and g(x) 8x2 7x
74. f(x) 5x2 7x and g(x) 9x2 3
Find values for a, b, c, and d so that the following equations are true.
73. (a) 3x2 7x 5;
(b) 1; (c) 1
2
74. (a) 4x 7x 3;
(b) 8; (c) 8
75. a 3; b 5; c 0;
d 1
76. a 1; b 4; c 2;
d6
77. 28x 4
78. 12x 4
79. x2 65x 150
80. 2y2 3y 8
75. 3ax4 5x3 x2 cx 2 9x4 bx3 x2 2d
76. (4ax3 3bx2 10) 3(x3 4x2 cx d) x3 6x 8
77. Geometry. A rectangle has sides of 8x 9 and 6x 7. Find the polynomial that
represents its perimeter.
78. Geometry. A triangle has sides 3x 7, 4x 9, and 5x 6. Find the polynomial
that represents its perimeter.
79. Business. The cost of producing x units of an item is C 150 25x. The revenue for selling x units is R 90x x2. The profit is given by the revenue minus
the cost. Find the polynomial that represents profit.
80. Business. The revenue for selling y units is R 3y2 2y 5 and the cost of producing y units is y2 y 3. Find the polynomial that represents profit.