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Holt Algebra Adding and Subtracting Polynomials 7-6 Adding and Subtracting Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Algebra Adding and Subtracting Polynomials Warm Up Simplify each expression by combining like terms. 1. 4x + 2x 2. 3y + 7y 3. 8p – 5p 4. 5n + 6n 2 Simplify each expression. 5. 3(x + 4) 6. –2(t + 3) 7. –1(x 2 – 4x – 6) 6x6x 10y 3p3p not like terms 3x + 12 –2t – 6 –x 2 + 4x + 6

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Holt Algebra Adding and Subtracting Polynomials Add and subtract polynomials. Objective

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Holt Algebra Adding and Subtracting Polynomials Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Add or Subtract.. Example 1: Adding and Subtracting Monomials A. 12p p 2 + 8p 3 12p p 2 + 8p 3 12p 3 + 8p p 2 20p p 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. B. 5x 2 – 6 – 3x + 8 5x 2 – 6 – 3x + 8 5x 2 – 3x + 8 – 6 5x 2 – 3x + 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Add or Subtract.. Example 1: Adding and Subtracting Monomials C. t 2 + 2s 2 – 4t 2 – s 2 t 2 – 4t 2 + 2s 2 – s 2 t 2 + 2s 2 – 4t 2 – s 2 –3t 2 + s 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. D. 10m 2 n + 4m 2 n – 8m 2 n 10m 2 n + 4m 2 n – 8m 2 n 6m2n6m2n Identify like terms. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7. Remember!

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Holt Algebra Adding and Subtracting Polynomials Check It Out! Example 1 a. 2s 2 + 3s 2 + s Add or subtract. 2s 2 + 3s 2 + s 5s 2 + s b. 4z 4 – z z 4 – z z z 4 – z 4 – 6 Identify like terms. Combine like terms. Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Check It Out! Example 1 c. 2x 8 + 7y 8 – x 8 – y 8 Add or subtract. 2x 8 + 7y 8 – x 8 – y 8 2x 8 – x 8 + 7y 8 – y 8 x 8 + 6y 8 d. 9b 3 c 2 + 5b 3 c 2 – 13b 3 c 2 9b 3 c 2 + 5b 3 c 2 – 13b 3 c 2 b3c2 b3c2 Identify like terms. Combine like terms. Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5x 2 + 4x + 1) + (2x 2 + 5x + 2) = (5x 2 + 2x 2 ) + (4x + 5x) + (1 + 2) = 7x 2 + 9x + 3 5x 2 + 4x x 2 + 5x + 2 7x2 + 9x + 37x2 + 9x + 3

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Holt Algebra Adding and Subtracting Polynomials Add. Example 2: Adding Polynomials A. (4m 2 + 5) + (m 2 – m + 6) (4m 2 + 5) + (m 2 – m + 6) (4m 2 + m 2 ) + (–m) +(5 + 6) 5m 2 – m + 11 Identify like terms. Group like terms together. Combine like terms. B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) (10xy – 3xy) + x + y 7xy + x + y Identify like terms. Group like terms together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Add. Example 2C: Adding Polynomials (6x 2 – 4y) + (3x 2 + 3y – 8x 2 – 2y) Identify like terms. Group like terms together within each polynomial. Combine like terms. (6x 2 – 4y) + (3x 2 + 3y – 8x 2 – 2y) (6x 2 + 3x 2 – 8x 2 ) + (3y – 4y – 2y) Use the vertical method. 6x 2 – 4y + –5x 2 + y x 2 – 3y Simplify.

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Holt Algebra Adding and Subtracting Polynomials Add. Example 2D: Adding Polynomials Identify like terms. Group like terms together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Check It Out! Example 2 Add (5a 3 + 3a 2 – 6a + 12a 2 ) + (7a 3 – 10a). (5a 3 + 3a 2 – 6a + 12a 2 ) + (7a 3 – 10a) (5a 3 + 7a 3 ) + (3a a 2 ) + (–10a – 6a) 12a a 2 – 16a Identify like terms. Group like terms together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: –(2x 3 – 3x + 7)= –2x 3 + 3x – 7

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Holt Algebra Adding and Subtracting Polynomials Subtract. Example 3A: Subtracting Polynomials (x 3 + 4y) – (2x 3 ) (x 3 + 4y) + (–2x 3 ) (x 3 – 2x 3 ) + 4y –x 3 + 4y Rewrite subtraction as addition of the opposite. Identify like terms. Group like terms together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Subtract. Example 3B: Subtracting Polynomials (7m 4 – 2m 2 ) – (5m 4 – 5m 2 + 8) (7m 4 – 2m 2 ) + (–5m 4 + 5m 2 – 8) (7m 4 – 5m 4 ) + (–2m 2 + 5m 2 ) – 8 (7m 4 – 2m 2 ) + (–5m 4 + 5m 2 – 8) 2m 4 + 3m 2 – 8 Rewrite subtraction as addition of the opposite. Identify like terms. Group like terms together. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Subtract. Example 3C: Subtracting Polynomials (–10x 2 – 3x + 7) – (x 2 – 9) (–10x 2 – 3x + 7) + (–x 2 + 9) –10x 2 – 3x + 7 –x 2 + 0x + 9 –11x 2 – 3x + 16 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0x as a placeholder. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Subtract. Example 3D: Subtracting Polynomials (9q 2 – 3q) – (q 2 – 5) (9q 2 – 3q) + (–q 2 + 5) 9q 2 – 3q q 2 – 0q + 5 8q 2 – 3q + 5 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0 and 0q as placeholders. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials Check It Out! Example 3 Subtract. (2x 2 – 3x 2 + 1) – (x 2 + x + 1) (2x 2 – 3x 2 + 1) + (–x 2 – x – 1) –x 2 + 0x –x 2 – x – 1 –2x 2 – x Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0x as a placeholder. Combine like terms.

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Holt Algebra Adding and Subtracting Polynomials A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x 2 + 7x – 5 and the area of plot B can be represented by 5x 2 – 4x Write a polynomial that represents the total area of both plots of land. Example 4: Application (3x 2 + 7x – 5) (5x 2 – 4x + 11) 8x 2 + 3x + 6 Plot A. Plot B. Combine like terms. +

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Holt Algebra Adding and Subtracting Polynomials Check It Out! Example 4 The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant. Use the information above to write a polynomial that represents the total profits from both plants. –0.03x x – 1500 Eastern plant profit. –0.02x x – 1700 Southern plant profit. Combine like terms. + –0.05x x – 3200

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Holt Algebra Adding and Subtracting Polynomials Lesson Quiz: Part I Add or subtract. 1. 7m 2 + 3m + 4m 2 2. (r 2 + s 2 ) – (5r 2 + 4s 2 ) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d 2 – 8) + (6d 2 – 2d +1) (–4r 2 – 3s 2 ) 11m 2 + 3m 18pq – 2p 20d 2 – 2d – 7 5. (2.5ab + 14b) – (–1.5ab + 4b)4ab + 10b

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Holt Algebra Adding and Subtracting Polynomials Lesson Quiz: Part II 6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x x + 9, and the area of the second wall is modeled by 36x 2 – 12x + 1. Write a polynomial that represents the total area of the two walls. 40x

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