4 4 ADDITION AND SUBTRACTION OF POLYNOMIALS a

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. Copyright © 2015, 2011, and 2007 Pearson Education, Inc.

Example Add. ( 6 x 3 + 7 x 2) + (5 x 3 + 4 x 2 + 3) Solution ( 6 x 3 + 7 x 2) + (5 x 3 + 4 x 2 + 3) = ( 6 + 5)x 3 + 4 x 2 + 7 x + ( 2 + 3) = x 3 + 4 x 2 + 7 x + 1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2

Example Add: (3 4 x + 2 x 2) + ( 6 + 8 x 4 x 2 + 2 x 3) Solution (3 4 x + 2 x 2) + ( 6 + 8 x 4 x 2 + 2 x 3) = (3 6) + ( 4 + 8)x + (2 4)x 2 + 2 x 3 = 3 + 4 x 2 x 2 + 2 x 3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 3

Example Add: 10 x 5 3 x 3 + 7 x 2 + 4 and 6 x 4 8 x 2 + 7 and 4 x 6 6 x 5 + 2 x 2 + 6 Solution 10 x 5 3 x 3 + 7 x 2 + 4 6 x 4 8 x 2 + 7 4 x 6 6 x 5 + 2 x 2 + 6 4 x 6 + 4 x 5 + 6 x 4 3 x 3 + x 2 + 17 The answer is 4 x 6 + 4 x 5 + 6 x 4 3 x 3 + x 2 + 17. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 4

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 © 2015, 2011, and 2007 Pearson Education, Inc. 5

Example Simplify: ( 8 x 4 x 3 + 9 x 2 2 x + 72) Solution ( 8 x 4 x 3 + 9 x 2 2 x + 72) = 8 x 4 + x 3 9 x 2 + 2 x 72 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 6

Subtraction of Polynomials We can now subtract one polynomial from another by adding the opposite of the polynomial being subtracted. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 7

Example (10 x 5 + 2 x 3 3 x 2 + 5) ( 3 x 5 + 2 x 4 5 x 3 4 x 2) Solution (10 x 5 + 2 x 3 3 x 2 + 5) ( 3 x 5 + 2 x 4 5 x 3 4 x 2) = 10 x 5 + 2 x 3 3 x 2 + 5 + 3 x 5 2 x 4 + 5 x 3 + 4 x 2 = 13 x 5 2 x 4 + 7 x 3 + x 2 + 5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 8

Example Subtract: (8 x 5 + 2 x 3 10 x) (4 x 5 5 x 3 + 6) Solution (8 x 5 + 2 x 3 10 x) (4 x 5 5 x 3 + 6) = 8 x 5 + 2 x 3 10 x + ( 4 x 5) + 5 x 3 6 = 4 x 5 + 7 x 3 10 x 6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9

Example Write in columns and subtract: (6 x 2 4 x + 7) (10 x 2 6 x 4) Solution 6 x 2 4 x + 7 (10 x 2 6 x 4) 4 x 2 + 2 x + 11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Remember to change the signs 10

Example 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 5 ft 7 ft w ft Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 11

continued 2. Translate. Rewording: Area of Area deck tub = left over Translating: w ft 5 ft 7 ft = Area left over 3. Carry out. w 2 ft 2 35 ft 2 = 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 (w 2 35)ft 2. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 12