This document discusses adding, subtracting, and simplifying polynomials. It provides examples of adding polynomials by combining like terms. It also demonstrates subtracting polynomials by adding the opposite of the polynomial being subtracted. Finally, it gives an example of using polynomials to represent the perimeter and area of a deck with a hot tub installed on it.

1. ADDITION AND SUBTRACTION OF
4.4 POLYNOMIALS
a Add polynomials.
b Simplify the opposite of a polynomial.
c Subtract polynomials.
d Use polynomials to represent perimeter and
area.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Objective a
Add polynomials.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2

3. Example A
Add. (6x3 + 7x  2) + (5x3 + 4x2 + 3)
Solution
(6x3 + 7x  2) + (5x3 + 4x2 + 3)
= (6 + 5)x3 + 4x2 + 7x + (2 + 3)
= x3 + 4x2 + 7x + 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3

4. Example B
Add: (3  4x + 2x2) + (6 + 8x  4x2 + 2x3)
Solution
(3  4x + 2x2) + (6 + 8x  4x2 + 2x3)
= (3  6) + (4 + 8)x + (2  4)x2 + 2x3
= 3 + 4x  2x2 + 2x3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4

5. Example C
Add: 10x5  3x3 + 7x2 + 4 and 6x4  8x2 + 7 and
4x6  6x5 + 2x2 + 6
Solution
10x5  3x3 + 7x2 + 4
6x4  8x2 + 7
4x6  6x5 + 2x2 + 6
4x6 + 4x5 + 6x4  3x3 + x2 + 17
The answer is 4x6 + 4x5 + 6x4  3x3 + x2 + 17.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5

6. Objective b
Simplify the opposite of a polynomial.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6

7. Opposites of Polynomials
To find an equivalent polynomial for the
opposite, or additive inverse, of a polynomial,
change the sign of every term. This is the same
as multiplying by 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7

8. Example D
Simplify: (8x4  3 x3 + 9x2  2x + 72)
4
Solution
(8x 4
4  3 x3 + 9x2  2x + 72)
 9x2 + 2x  72
3 3
= 8x4 + 4x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8

9. Objective c
Subtract polynomials.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9

10. Subtraction of Polynomials
We can now subtract one polynomial from another by
adding the opposite of the polynomial being
subtracted.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10

11. Example E Subtract:
(10x5 + 2x3  3x2 + 5)  (3x5 + 2x4  5x3  4x2)
Solution
(10x5 + 2x3  3x2 + 5)  (3x5 + 2x4  5x3  4x2)
= 10x5 + 2x3  3x2 + 5 + 3x5  2x4 + 5x3 + 4x2
= 13x5  2x4 + 7x3 + x2 + 5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11

12. Example F
Subtract: (8x5 + 2x3  10x)  (4x5  5x3 + 6)
Solution
(8x5 + 2x3  10x)  (4x5  5x3 + 6)
= 8x5 + 2x3  10x + (4x5) + 5x3  6
= 4x5 + 7x3  10x  6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12

13. Example G
Write in columns and subtract:
(6x2  4x + 7)  (10x2  6x  4)
Solution
6x2  4x + 7
(10x2  6x  4) Remember to change
the signs
4x2 + 2x + 11
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13

14. Objective d
Use polynomials to represent
perimeter and area.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14

15. Example H
A 6-ft by 5-ft hot tub is installed on an outdoor
deck measuring w ft by w ft. Find a polynomial
for the remaining area of the deck.
Solution
1. Familiarize. We make a drawing of the
situation as follows.
5 ft w ft
7 ft
w ft
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15

16. continued
2. Translate.
Rewording: Area of Area of Area
deck  tub = left over
Translating: w ft  w ft  5 ft  7 ft = Area left over
3. Carry out.
w2 ft2  35 ft2 = Area left over.
4. Check. As a partial check, note that the units in the
answer are square feet, a measure of area, as
expected.
5. State. The remaining area in the yard is (w2  35)ft2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16

17. 1. Add: (6x5 – x3 + 4x2 + x) +
(x4 – x3 + 8x2 – x + 2).
a) 7x5 – 2x3 + 12x2 + 2x + 2
b) 6x5 + x4 – 2x3 + 12x2 + 2
c) 7x5 – 2x3 + 12x2 + 2
d) 6x5 + x4 + 12x2 + 2
Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17

18. 1. Add: (6x5 – x3 + 4x2 + x) +
(x4 – x3 + 8x2 – x + 2).
a) 7x5 – 2x3 + 12x2 + 2x + 2
b) 6x5 + x4 – 2x3 + 12x2 + 2
c) 7x5 – 2x3 + 12x2 + 2
d) 6x5 + x4 + 12x2 + 2
Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18

19. 2. Subtract: (1.1x3 – 8x2 + 4.5x) –
(3x3 – x2 + 0.5x – 6).
a) –2.2x3 – 7x2 + 4x + 6
b) –1.9x3 – 7x2 + 4x + 6
c) –2.9x3 – 7x2 + 4x + 6
d) –1.9x3 – 9x2 + 4x + 6
Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19

20. 2. Subtract: (1.1x3 – 8x2 + 4.5x) –
(3x3 – x2 + 0.5x – 6).
a) –2.2x3 – 7x2 + 4x + 6
b) –1.9x3 – 7x2 + 4x + 6
c) –2.9x3 – 7x2 + 4x + 6
d) –1.9x3 – 9x2 + 4x + 6
Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20