1. 13-4 Subtracting Polynomials Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation Pre-Algebra

2. Warm Up Write the opposite of each integer. 1. 10 Subtract. 3. 19 – (–12) Add. 5. (3x 2 + 7) + (x 2 – 3x) 6. (2m 2 – 3m) + (–5m 2 + 2) –10 31 –37 13-4 Subtracting Polynomials 2. –7 7 4. –16 – 21 4x 2 – 3x + 7 –3m 2 – 3m + 2 Pre-Algebra

3. Problem of the Day Tara has 4 pairs of shorts, 3 tops, and 2 pairs of sandals. If she wants to wear a completely different outfit than she wore yesterday, how many combinations does she have to choose from? 6 Pre-Algebra 13-4 Subtracting Polynomials

4. Learn to subtract polynomials. Pre-Algebra 13-4 Subtracting Polynomials

5. Subtraction is the opposite of addition. To subtract a polynomial, you need to find its opposite. Pre-Algebra 13-4 Subtracting Polynomials

6. Find the opposite of each polynomial. Additional Example 1A & 1B: Finding the Opposite of a Polynomial A. 8x 3 y 4 z 2 – (8x 3 y 4 z 2 ) –8x3y4z2–8x3y4z2 The opposite of a is –a. B. – 3x 4 + 8x 2 – ( – 3x 4 + 8x 2 ) 3x 4 – 8x 2 Distribute the sign. Pre-Algebra 13-4 Subtracting Polynomials

7. Find the opposite of the polynomial. Additional Example 1C: Finding the Opposite of a Polynomial A. 9a 6 b 4 + a 4 b 2 – 1 – (9a 6 b 4 + a 4 b 2 – 1) – 9a 6 b 4 – a 4 b 2 + 1 Distribute the sign. Pre-Algebra 13-4 Subtracting Polynomials

8. Try This: Example 1A & 1B Insert Lesson Title Here Find the opposite of each polynomial. A. 4d 2 e 3 f 3 – (4d 2 e 3 f 3 ) The opposite of a is –a. B. – 4a 2 + 4a 4 – ( – 4a 2 + 4a 4 ) 4a 2 – 4a 4 Distribute the sign. –4d2e3f3–4d2e3f3 Pre-Algebra 13-4 Subtracting Polynomials

9. Try This: Example 1C Insert Lesson Title Here Find the opposite of the polynomial. A. 9a 6 b 4 + a 4 b 2 – 1 – (9a 6 b 4 + a 4 b 2 – 1) – 9a 6 b 4 – a 4 b 2 + 1 Distribute the sign. Pre-Algebra 13-4 Subtracting Polynomials

10. To subtract a polynomial, add its opposite. Pre-Algebra 13-4 Subtracting Polynomials

11. Subtract. Additional Example 2A: Subtracting Polynomials Horizontally A. (5x 2 + 2x – 3) – (3x 2 + 8x – 4) = (5x 2 + 2x – 3) + ( – 3x 2 – 8x + 4) = 5x 2 + 2x – 3 – 3x 2 – 8x + 4 Add the opposite. Associative property. = 2x 2 – 6x + 1 Combine like terms. Pre-Algebra 13-4 Subtracting Polynomials

12. Subtract. Additional Example 2B: Subtracting Polynomials Horizontally B. (b 2 + 4b – 1) – (7b 2 – b – 1) = (b 2 + 4b – 1) + (–7b 2 + b + 1) = b 2 + 4b – 1 – 7b 2 + b + 1 = – 6b 2 + 5b Add the opposite. Associative property. Combine like terms. Pre-Algebra 13-4 Subtracting Polynomials

13. Try This: Example 2A Insert Lesson Title Here Subtract. A. (2y 3 + 3y + 5) – (4y 3 + 3y + 5) Add the opposite. Associative property. = – 2y 3 Combine like terms. = (2y 3 + 3y + 5) + (–4y 3 – 3y – 5) = 2y 3 + 3y + 5 – 4y 3 – 3y – 5 Pre-Algebra 13-4 Subtracting Polynomials

14. Try This: Example 2B Insert Lesson Title Here Subtract. B. (c 3 + 2c 2 + 3) – (4c 3 – c 2 – 1) = – 3c 3 + 3c 2 + 4 Add the opposite. Associative property. Combine like terms. = (c 3 + 2c 2 + 3) + (–4c 3 + c 2 + 1) = c 3 + 2c 2 + 3 – 4c 3 + c 2 + 1 Pre-Algebra 13-4 Subtracting Polynomials

15. You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the decimal points. Pre-Algebra 13-4 Subtracting Polynomials

16. Subtract. Additional Example 3A: Subtracting Polynomials Vertically A. (2n 2 – 4n + 9) – (6n 2 – 7n + 5) (2n 2 – 4n + 9)2n 2 – 4n + 9 – (6n 2 – 7n + 5) + – 6n 2 + 7n – 5 – 4n 2 + 3n + 4 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

17. Subtract. Additional Example 3B: Subtracting Polynomials Vertically B. (10x 2 + 2x – 7) – (x 2 + 5x + 1) (10x 2 + 2x – 7)10x 2 + 2x – 7 – (x 2 + 5x + 1)+ – x 2 – 5x – 1 9x 2 – 3x – 8 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

18. Subtract. Additional Example 3C: Subtracting Polynomials Vertically C. (6a 4 – 3a 2 – 8) – ( – 2a 4 + 7) (6a 4 – 3a 2 – 8) 6a 4 – 3a 2 – 8 – ( – 2a 4 + 7) + 2a 4 – 7 8a 4 – 3a 2 – 15 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

19. Try This: Example 3A Subtract. Insert Lesson Title Here A. (4r 3 + 4r + 6) – (6r 3 + 3r + 3) (4r 3 + 4r + 6)4r 3 + 4r + 6 – (6r 3 + 3r + 3)+ – 6r 3 – 3r – 3 – 2r 3 + r + 3 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

20. Try This: Example 3B Insert Lesson Title Here Subtract. B. (13y 2 – 2x + 5) – (y 2 + 5x – 9) (13y 2 – 2x + 5)13y 2 – 2x + 5 – (y 2 + 5x – 9) + – y 2 – 5x + 9 12x 2 – 7x + 14 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

21. Try This: Example 3C Insert Lesson Title Here Subtract. C. (5x 2 + 2x + 5) – ( – 3x 2 – 7x) (5x 2 + 2x + 5) 5x 2 + 2x + 5 – ( – 3x 2 – 7x) +3x 2 + 7x 8x 2 + 9x + 5 Add the opposite. Pre-Algebra 13-4 Subtracting Polynomials

22. Suppose the cost in dollars of producing x bookcases is given by the polynomial 250 + 128x, and the revenue generated from sales is given by the polynomial 216x – 75. Find a polynomial expression for the profit from producing and selling x bookcases, and evaluate the expression for x = 95. Additional Example 4: Business Application 216x – 75 – (250 + 128x)revenue – cost 216x – 75 + ( – 250 – 128x) Add the opposite. 216x – 75 – 250 – 128x Associative Property Pre-Algebra 88x – 325 Combine like terms. 13-4 Subtracting Polynomials

23. Additional Example 4 Continued The profit is given by the polynomial 88x – 325. 88(95) – 325 = 8360 – 325 = 8035 The profit is $8035. For x = 95, Pre-Algebra 13-4 Subtracting Polynomials

24. Try This: Example 4 Insert Lesson Title Here Suppose the cost in dollars of producing x baseball bats is given by the polynomial 6 + 12x, and the revenue generated from sales is given by the polynomial 35x – 5. Find a polynomial expression for the profit from producing and selling x baseball bats, and evaluate the expression for x = 50. 35x – 5 – (6 + 12x)revenue – cost 35x – 5 + ( – 6 – 12x) Add the opposite. 35x – 5 – 6 – 12x Associative Property Pre-Algebra 23x – 11 Combine like terms. 13-4 Subtracting Polynomials

25. Try This: Example 4 Continued Insert Lesson Title Here The profit is given by the polynomial 23x – 11. 23(50) – 11 = 1150 – 11 = 1139 The profit is $1139. For x = 50, Pre-Algebra 13-4 Subtracting Polynomials

26. Lesson Quiz Find the opposite of each polynomial. Subtract. 3. (3z 2 – 7z + 6) – (2z 2 + z – 12) 3m 3 – 2m 2 n Insert Lesson Title Here z 2 – 8z + 18 –3a2b2c3–3a2b2c3 2. – 3m 3 + 2m 2 n Pre-Algebra 4. – 18h 3 – (4h 3 + h 2 – 12h + 2) 5. (3b 2 c + 5bc 2 – 8b 2 ) – (4b 2 c + 2bc 2 – c 2 ) – b 2 c + 3bc 2 – 8b 2 + c 2 – 22h 3 – h 2 + 12h – 2 13-4 Subtracting Polynomials 1. 3a 2 b 2 c 3