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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up Write the opposite of each integer. 1. 10 Subtract. 3. 19 – ( – 12) Add. 5. (3 x 2 + 7) + ( x 2 – 3 x ) 6. (2 m 2 – 3 m ) + ( – 5 m 2 + 2). –10. 2. –7. 7. 31. 4. –16 – 21. –37. 4 x 2 – 3 x + 7.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Write the opposite of each integer. 1.10 Subtract. 3. 19 – (–12) Add. 5. (3x2 + 7) + (x2– 3x) 6. (2m2 – 3m) + (–5m2 + 2) –10 2. –7 7 31 4. –16 – 21 –37 4x2– 3x + 7 –3m2– 3m + 2

  3. California Standards 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

  4. Recall that to subtract an integer, you add its opposite. To subtract a polynomial, you first need to find its opposite.

  5. Additional Example 1: Finding the Opposite of a Polynomial Find the opposite of each polynomial. A. 8x3y4z2 –(8x3y4z2) The opposite of a positive term is a negative term. –8x3y4z2 B. –3x4 + 8x2 –(–3x4 + 8x2) Distributive Property. 3x4– 8x2

  6. Additional Example 1: Finding the Opposite of a Polynomial Find the opposite of the polynomial. C. 9a6b4 + a4b2– 1 –(9a6b4 + a4b2– 1) Distributive Property –9a6b4 –a4b2 + 1

  7. Check It Out! Example 1 Find the opposite of each polynomial. A. 4d2e3f3 –(4d2e3f3) The opposite of a positive term is a negative term. –4d2e3f3 B. –4a2 + 4a4 –(–4a2 + 4a4) Distributive Property 4a2– 4a4

  8. Check It Out! Example 1 Find the opposite of the polynomial. C. 9a6b4 + a4b2– 1 –(9a6b4 + a4b2– 1) –9a6b4 –a4b2 + 1 Distributive Property

  9. To subtract a polynomial, add its opposite.

  10. Additional Example 2: Subtracting Polynomials Horizontally Subtract. A. (5x2 + 2x– 3) – (3x2 + 8x– 4) Add the opposite. (5x2 + 2x– 3) + (–3x2– 8x+ 4) Commutative Property 5x2 – 3x2 + 2x– 8x – 3+ 4 2x2– 6x + 1 Combine like terms.

  11. Additional Example 2: Subtracting Polynomials Horizontally Subtract. B. (b2 + 4b – 1) – (7b2–b– 1) Add the opposite. (b2 + 4b – 1) + (–7b2+b+ 1) Commutative Property b2 – 7b2 + 4b + b – 1+ 1 –6b2 + 5b Combine like terms.

  12. Check It Out! Example 2 Subtract. A. (2y3 + 3y + 5) – (4y3 + 3y + 5) Add the opposite. (2y3 + 3y + 5) + (–4y3– 3y – 5) Commutative Property 2y3 – 4y3 + 3y– 3y + 5 – 5 –2y3 Combine like terms.

  13. Check It Out! Example 2B Subtract. (c3 + 2c2+ 3) – (4c3–c2– 1) (c3 + 2c2+ 3) + (–4c3+c2+ 1) Add the opposite. Commutative Property c3 – 4c3 + 2c2+ c2+ 3 + 1 –3c3 + 3c2 + 4 Combine like terms.

  14. You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms.

  15. Additional Example 3: Subtracting Polynomials Vertically Subtract. A. (2n2– 4n + 9) – (6n2– 7n + 5) (2n2– 4n + 9) 2n2– 4n + 9 – (6n2 – 7n + 5) +–6n2 + 7n –5 Add the opposite. –4n2 + 3n + 4

  16. Additional Example 3: Subtracting Polynomials Vertically Subtract. B. (10x2 + 2x –7) – (x2 + 5x + 1) (10x2 + 2x –7) 10x2 + 2x –7 Add the opposite. – (x2 + 5x + 1) + –x2– 5x– 1 9x2– 3x– 8

  17. Additional Example 3: Subtracting Polynomials Vertically Subtract. C. (6a4– 3a2–8) – (–2a4 + 7) (6a4– 3a2–8) 6a4– 3a2–8 – (–2a4 + 7) + 2a4– 7 Rearrange terms as needed. 8a4 – 3a2– 15

  18. Check It Out! Example 3 Subtract. A. (4r3 + 4r + 6) – (6r3 + 3r + 3) (4r3 + 4r + 6) 4r3 + 4r + 6 Add the opposite. – (6r3 + 3r + 3) + –6r3– 3r –3 –2r3 + r + 3

  19. Check It Out! Example 3 Subtract. B. (13y2– 2x + 5) – (y2 + 5x– 9) (13y2– 2x + 5) 13y2– 2x + 5 – (y2 + 5x– 9) + –y2– 5x + 9 Add the opposite. 12y2– 7x + 14

  20. Check It Out! Example 3 Subtract. C. (5x2 + 2x + 5) – (–3x2– 7x) (5x2 + 2x + 5) 5x2 + 2x + 5 +3x2 + 7x – (–3x2– 7x) Add the opposite. 8x2 + 9x+ 5

  21. Additional Example 4: Business Application 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. 216x – 75 – (250 + 128x) revenue – cost Add the opposite. 216x – 75 + (–250 – 128x) Commutative Property 216x – 128x – 75 – 250 Combine like terms. 88x – 325

  22. Additional Example 4 Continued The profit is given by the polynomial 88x– 325. For x = 95, 88(95) – 325 = 8360 – 325 = 8035 The profit is $8035.

  23. 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 Add the opposite. 35x – 5 + (–6 – 12x) Commutative Property 35x – 12x – 5 – 6 23x – 11 Combine like terms.

  24. Check It Out! Example 4 Continued The profit is given by the polynomial 23x– 11. For x = 50, 23(50) – 11 = 1150 – 11 = 1139 The profit is $1139.

  25. Lesson Quiz Find the opposite of each polynomial. Subtract. 3. (3z2 – 7z + 6) – (2z2 + z– 12) 2.–3m3 + 2m2n 3m3– 2m2n 1. 3a2b2c3 –3a2b2c3 z2– 8z + 18 4.–18h3– (4h3 + h2– 12h + 2) 5. (3b2c + 5bc2– 8b2) – (4b2c + 2bc2–c2) –22h3–h2 + 12h– 2 –b2c + 3bc2– 8b2 + c2

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