1. Holt CA Course 1 12-4Subtracting Polynomials Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

2. Holt CA Course 1 12-4Subtracting Polynomials 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 2. –7 7 4. –16 – 21 4x 2 – 3x + 7 –3m 2 – 3m + 2

3. Holt CA Course 1 12-4Subtracting Polynomials Preview of Algebra 1 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Also covered: 7AF1.3 California Standards

4. Holt CA Course 1 12-4Subtracting Polynomials Recall that to subtract an integer, you add its opposite. To subtract a polynomial, you first need to find its opposite.

5. Holt CA Course 1 12-4Subtracting Polynomials Find the opposite of each polynomial. Additional Example 1: 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 positive term is a negative term. B. – 3x 4 + 8x 2 – ( – 3x 4 + 8x 2 ) 3x 4 – 8x 2 Distributive Property.

6. Holt CA Course 1 12-4Subtracting Polynomials Find the opposite of the polynomial. Additional Example 1: Finding the Opposite of a Polynomial C. 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 Distributive Property

7. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 1 Find the opposite of each polynomial. A. 4d 2 e 3 f 3 – (4d 2 e 3 f 3 ) B. – 4a 2 + 4a 4 – ( – 4a 2 + 4a 4 ) 4a 2 – 4a 4 Distributive Property –4d2e3f3–4d2e3f3 The opposite of a positive term is a negative term.

8. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 1 Find the opposite of the polynomial. C. 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 Distributive Property

9. Holt CA Course 1 12-4Subtracting Polynomials To subtract a polynomial, add its opposite.

10. Holt CA Course 1 12-4Subtracting Polynomials Subtract. Additional Example 2: Subtracting Polynomials Horizontally A. (5x 2 + 2x – 3) – (3x 2 + 8x – 4) (5x 2 + 2x – 3) + ( – 3x 2 – 8x + 4) 5x 2 – 3x 2 + 2x – 8x – 3 + 4 Add the opposite. Commutative Property 2x 2 – 6x + 1 Combine like terms.

11. Holt CA Course 1 12-4Subtracting Polynomials Subtract. Additional Example 2: Subtracting Polynomials Horizontally B. (b 2 + 4b – 1) – (7b 2 – b – 1) (b 2 + 4b – 1) + (–7b 2 + b + 1) b 2 – 7b 2 + 4b + b – 1 + 1 – 6b 2 + 5b Add the opposite. Commutative Property Combine like terms.

12. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 2 Subtract. A. (2y 3 + 3y + 5) – (4y 3 + 3y + 5) Add the opposite. Commutative Property –2y3–2y3 Combine like terms. (2y 3 + 3y + 5) + (–4y 3 – 3y – 5) 2y 3 – 4y 3 + 3y – 3y + 5 – 5

13. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 2B Subtract. (c 3 + 2c 2 + 3) – (4c 3 – c 2 – 1) – 3c 3 + 3c 2 + 4 Add the opposite. Commutative Property Combine like terms. (c 3 + 2c 2 + 3) + (–4c 3 + c 2 + 1) c 3 – 4c 3 + 2c 2 + c 2 + 3 + 1

14. Holt CA Course 1 12-4Subtracting Polynomials You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms.

15. Holt CA Course 1 12-4Subtracting Polynomials Subtract. Additional Example 3: 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.

16. Holt CA Course 1 12-4Subtracting Polynomials Subtract. Additional Example 3: 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.

17. Holt CA Course 1 12-4Subtracting Polynomials Subtract. Additional Example 3: 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 Rearrange terms as needed.

18. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 3 Subtract. 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.

19. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 3 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 12y 2 – 7x + 14 Add the opposite.

20. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 3 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.

21. Holt CA Course 1 12-4Subtracting Polynomials 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 – 128x – 75 – 250 Commutative Property 88x – 325 Combine like terms.

22. Holt CA Course 1 12-4Subtracting Polynomials 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,

23. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 4 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 – 12x – 5 – 6 Commutative Property 23x – 11 Combine like terms.

24. Holt CA Course 1 12-4Subtracting Polynomials Check It Out! Example 4 Continued The profit is given by the polynomial 23x – 11. 23(50) – 11 = 1150 – 11 = 1139 The profit is $1139. For x = 50,

25. Holt CA Course 1 12-4Subtracting Polynomials Lesson Quiz Find the opposite of each polynomial. Subtract. 3. (3z 2 – 7z + 6) – (2z 2 + z – 12) 3m 3 – 2m 2 n z 2 – 8z + 18 –3a2b2c3–3a2b2c3 2. – 3m 3 + 2m 2 n 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 1. 3a 2 b 2 c 3