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Warm Up Combine like terms. 1. 9 x + 4 x 2. –3 y + 7 y 3. 7 n + (–8 n ) + 12 n

7 x 2 – 3 x + 1. Warm Up Combine like terms. 1. 9 x + 4 x 2. –3 y + 7 y 3. 7 n + (–8 n ) + 12 n Find the perimeter of each rectangle. 4. a 10 ft by 12 ft rectangle 5. a 5 m by 8 m rectangle Simplify. 6. 3(2 x 2 – x ) + x 2 + 1. 13 x. 4 y. 11 n. 44 ft. 26 m.

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Warm Up Combine like terms. 1. 9 x + 4 x 2. –3 y + 7 y 3. 7 n + (–8 n ) + 12 n

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  1. 7x2 – 3x + 1 Warm Up Combine like terms. 1.9x + 4x 2. –3y + 7y 3. 7n + (–8n) + 12n Find the perimeter of each rectangle. 4. a 10 ft by 12 ft rectangle 5. a 5 m by 8 m rectangle Simplify. 6. 3(2x2 – x) + x2+ 1 13x 4y 11n 44 ft 26 m

  2. Learn to add and subtract polynomials.

  3. 2 3 2 3 (5x + x + 2) + (4x + 6x ) 2 3 2 3 5x + x + 2 + 4x + 6x 2 3 9x + 7x + 2 Associative Property Combine like terms. Example 1A: Adding Polynomials Horizontally Add. (5x3 + x2 + 2) + (4x3 + 6x2)

  4. (6x3+ 8y2+ 5xy) + (4xy – 2y2) 6x3 + 8y2 + 5xy + 4xy – 2y2 2 3 6x + 6y + 9xy Associative Property Combine like terms. Example 1B: Adding Polynomials Horizontally Add. (6x3+ 8y2 + 5xy) + (4xy – 2y2)

  5. (3x2y – 5x) + (4x + 7) + 6x2y 3x2y – 5x + 4x + 7 + 6x2y 9x2y – x + 7 Associative Property Combine like terms. Example 1C: Adding Polynomials Horizontally Add. (3x2y – 5x) + (4x + 7) + 6x2y

  6. 2 4 2 4 (3y + y + 6) + (5y + 2y ) 2 4 2 4 3y + y + 6 + 5y + 2y 2 4 8y + 3y + 6 Associative Property Combine like terms. Example 2A Add. (3y4 + y2 + 6) + (5y4 + 2y2)

  7. 3 (9x + 6p2 + 3xy) + (8xy – 3p2) 2 2 3 9x + 6p + 3xy + 8xy – 3p 9x3+ 3p2 + 11xy Associative Property Combine like terms. Example 2B Add. (9x3 + 6p2 + 3xy) + (8xy – 3p2)

  8. (3z2w – 5x) + (2x + 8) + 6z2w 3z2w – 5x + 2x + 8 + 6z2w 9z2w – 3x + 8 Associative Property Combine like terms. Example 2C Add. (3z2w – 5x) + (2x + 8) + 6z2w

  9. You can also add polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. If the terms are rearranged, remember to keep the correct sign with each term.

  10. 4x2 + 2x + 11 + 2x2 + 6x + 9 6x2 + 8x + 20 Example 3: Adding Polynomials Vertically Add. A. (4x2 + 2x + 11) + (2x2 + 6x + 9) Place like terms in columns. Combine like terms.

  11. + 5mn2 + 2m – n 8mn2 – 4m + 5n 3mn2 – 6m + 6n –2y2 + 2 –x2y2+ 6x2 – 2y2 + 10 Example 3: Adding Polynomials Vertically Add. B. (3mn2 – 6m + 6n) + (5mn2 + 2m – n) C. (–x2y2 + 5x2) + (–2y2 + 2) + (x2 + 8) Place like terms in columns. Combine like terms. –x2y2 + 5x2 Place like terms in columns. + x2 + 8 Combine like terms.

  12. 6x2 + 6x + 13 + 3x2 + 2x + 4 9x2 + 8x + 17 Example 4 Add. A. (6x2 + 6x + 13) + (3x2+ 2x + 4) Place like terms in columns. Combine like terms.

  13. + 2mn2 – 2m – 2n 6mn2 + 4m 4mn2 + 6m + 2n 2y2 – 2 x2y2– 4x2 + 2y2 – 2 Example 4 Add. B. (4mn2 + 6m + 2n) + (2mn2 – 2m – 2n) C. (x2y2 – 5x2) + (2y2 – 2) + (x2) Place like terms in columns. Combine like terms. x2y2 – 5x2 Place like terms in columns. + x2 Combine like terms.

  14. Subtraction is the opposite of addition. To subtract a polynomial, you need to find its opposite.

  15. Example 1: Finding the Opposite of a Polynomial Find the opposite of each polynomial. A. 8x3y4z2 –(8x3y4z2) Distributive Property. –8x3y4z2 B. –3x4 + 8x2 –(–3x4 + 8x2) Distributive Property. 3x4– 8x2

  16. 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

  17. To subtract a polynomial, add its opposite.

  18. Example 1: Subtracting Polynomials Horizontally Subtract. A. (5x2 + 2x– 3) – (3x2 + 8x– 4) Add the opposite. = (5x2 + 2x– 3) + (–3x2– 8x+ 4) Associative property. = 5x2 + 2x– 3 – 3x2– 8x + 4 = 2x2– 6x + 1 Combine like terms.

  19. Example 1: Subtracting Polynomials Horizontally Subtract. B. (b2 + 4b – 1) – (7b2–b– 1) Add the opposite. = (b2 + 4b – 1) + (–7b2+b+ 1) Associative property. = b2 + 4b – 1 – 7b2 + b + 1 = –6b2 + 5b Combine like terms.

  20. Example 2A Subtract. (2y3 + 3y + 5) – (4y3 + 3y + 5) Add the opposite. = (2y3 + 3y + 5) + (–4y3– 3y – 5) Associative property. = 2y3 + 3y + 5 – 4y3– 3y– 5 = –2y3 Combine like terms.

  21. Example 2B Subtract. (c3 + 2c2+ 3) – (4c3–c2– 1) = (c3 + 2c2+ 3) + (–4c3+c2+ 1) Add the opposite. = c3 + 2c2+ 3 – 4c3 + c2 + 1 Associative property. = –3c3 + 3c2 + 4 Combine like terms.

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

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

  24. Example 4: Subtracting Polynomials Vertically Subtract. (10x2 + 2x –7) – (x2 + 5x + 1) (10x2 + 2x –7) 10x2 + 2x –7 Add the opposite. – (x2 + 5x + 1) + –x2– 5x– 1 9x2– 3x– 8

  25. Example 5: Subtracting Polynomials Vertically Subtract. (6a4– 3a2–8) – (–2a4 + 7) (6a4– 3a2–8) 6a4– 3a2–8 – (–2a4 + 7) + 2a4– 7 Rearrange as needed. 8a4 – 3a2– 15

  26. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  27. 9m2 – 3m + 6 3yz2+ 4yz + 7 2 7xy + 2x + 3y + 2 Lesson Quiz: Part I Add. 1. (2m2 – 3m + 7) + (7m2 – 1) 2. (yz2 + 5yz + 7) + (2yz2 – yz) 3. (2xy2 + 2x – 6) + (5xy2 + 3y + 8)

  28. 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

  29. Lesson Quiz for Student Response Systems 1. Add(4p2 – 8p +11) + (6p2 – 9). A. 10p2 – 8p + 2 B. 10p2 + 8p + 20 C. 2p2 + 8p + 2 D.10p2 – 8p + 20

  30. Lesson Quiz for Student Response Systems 2. Add(gh2 + 9gh + 11) + (3gh2 – gh). A. 2gh2 + 8gh + 11 B. 2gh2 + 10gh + 11 C. 4gh2 + 8gh + 11 D.4gh2 + 10gh + 11

  31. Lesson Quiz for Student Response Systems 3. Add(7uv3 + 11u) + (6uv3 – u –9) + (4u – 2). A. 13uv3 – 6u – 7 B. uv2 – 16u + 7 C. uv3 + 14u – 11 D.13uv3 + 14u – 11

  32. Lesson Quiz for Student Response Systems 3. Subtract. (11p2 –5p + 9) – (3p2 + p – 17) A. 14p2 +4p – 8 B. 14p2 –6p + 26 C. 8p2 –6p + 26 D.8p2 +4p – 8

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