1. ADDITION AND SUBTRACTION OF POLYNOMIALS
4.4
ADDITION AND SUBTRACTION OF POLYNOMIALS
a. Add polynomials.
b. Simplify the opposite of a polynomial.
c. Subtract polynomials.
d. Use polynomials to represent perimeter and area.

2. 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

3. 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

4. Add: 10x5  3x3 + 7x2 + 4 and 6x4  8x2 + 7 and 4x6  6x5 + 2x2 + 6
Solution
10x  3x3 + 7x2 + 4
6x  8x2 + 7
4x6  6x x2 + 6
4x6 + 4x5 + 6x4  3x3 + x2 + 17
The answer is 4x6 + 4x5 + 6x4  3x3 + x

5. 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.

6. Simplify: (8x4  x3 + 9x2  2x + 72)
Solution
(8x4  x3 + 9x2  2x + 72)
= 8x4 + x3  9x2 + 2x  72

7. Subtraction of Polynomials
We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted.

8. (10x5 + 2x3  3x2 + 5)  (3x5 + 2x4  5x3  4x2)
Solution
= 10x5 + 2x3  3x x5  2x4 + 5x3 + 4x2
= 13x5  2x4 + 7x3 + x2 + 5

9. Solution (8x5 + 2x3  10x)  (4x5  5x3 + 6)
Subtract: (8x5 + 2x3  10x)  (4x5  5x3 + 6)
Solution
(8x5 + 2x3  10x)  (4x5  5x3 + 6)
= 8x5 + 2x3  10x + (4x5) + 5x3  6
= 4x5 + 7x3  10x  6

10. Write in columns and subtract: (6x2  4x + 7)  (10x2  6x  4)
Solution
6x2  4x + 7
(10x2  6x  4)
4x2 + 2x + 11
Remember to change the signs

11. 1. Familiarize. We make a drawing of the situation as follows.
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.
w ft
7 ft
5 ft

12. 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.