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1. 1. ADDITION AND SUBTRACTION OF4.4 POLYNOMIALSa 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
3. 3. Example AAdd. (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. 4. Example BAdd: (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. 5. Example CAdd: 10x5  3x3 + 7x2 + 4 and 6x4  8x2 + 7 and4x6  6x5 + 2x2 + 6Solution 10x5  3x3 + 7x2 + 4 6x4  8x2 + 7 4x6  6x5 + 2x2 + 6 4x6 + 4x5 + 6x4  3x3 + x2 + 17The answer is 4x6 + 4x5 + 6x4  3x3 + x2 + 17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5
6. 6. Objective bSimplify the opposite of a polynomial. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6
7. 7. Opposites of PolynomialsTo find an equivalent polynomial for theopposite, or additive inverse, of a polynomial,change the sign of every term. This is the sameas multiplying by 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7
8. 8. Example DSimplify: (8x4  3 x3 + 9x2  2x + 72) 4Solution (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. 9. Objective cSubtract polynomials. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9
10. 10. Subtraction of PolynomialsWe can now subtract one polynomial from another byadding the opposite of the polynomial beingsubtracted. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10
11. 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. 12. Example FSubtract: (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. 13. Example GWrite 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. 14. Objective dUse polynomials to representperimeter and area. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14
15. 15. Example HA 6-ft by 5-ft hot tub is installed on an outdoordeck measuring w ft by w ft. Find a polynomialfor the remaining area of the deck.Solution1. 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. 16. continued2. Translate. Rewording: Area of Area of Area deck  tub = left overTranslating: w ft  w ft  5 ft  7 ft = Area left over3. 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. 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 + 2Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17
18. 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 + 2Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18
19. 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 + 6Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19
20. 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 + 6Section 4.4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20