7-6 Adding and Subtracting Polynomials
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1 Example 1: Adding and Subtracting Monomials Add or Subtract.. A. 12p p 2 + 8p 3 12p p 2 + 8p 3 12p 3 + 8p p 2 20p p 2 B. 5x 2 6 3x + 8 5x 2 6 3x + 8 5x 2 3x x 2 3x + 2 Rearrange terms so that like terms are together. Rearrange terms so that like terms are together.
2 Example 1: Adding and Subtracting Monomials Add or Subtract.. C. t 2 + 2s 2 4t 2 s 2 t 2 + 2s 2 4t 2 s 2 t 2 4t 2 + 2s 2 s 2 3t 2 + s 2 Rearrange terms so that like terms are together. D. 10m 2 n + 4m 2 n 8m 2 n 10m 2 n + 4m 2 n 8m 2 n 6m 2 n
3 Remember! 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.
4 Check It Out! Example 1 Add or subtract. a. 2s 2 + 3s 2 + s 2s 2 + 3s 2 + s 5s 2 + s b. 4z z z z z z z 4 6 Rearrange terms so that like terms are together.
5 Check It Out! Example 1 Add or subtract. c. 2x 8 + 7y 8 x 8 y 8 2x 8 + 7y 8 x 8 y 8 2x 8 x 8 + 7y 8 y 8 x 8 + 6y 8 Rearrange terms so that like terms are together. 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 b 3 c 2
6 Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: 5x 2 + 4x x 2 + 5x + 2 7x 2 + 9x + 3 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
7 Example 2: Adding Polynomials Add. 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 B. (10xy + x) + ( 3xy + y) (10xy + x) + ( 3xy + y) (10xy 3xy) + x + y 7xy + x + y Group like terms together. Group like terms together.
8 Example 2C: Adding Polynomials Add. (6x 2 4y) + (3x 2 + 3y 8x 2 2y) (6x 2 4y) + (3x 2 + 3y 8x 2 2y) (6x 2 + 3x 2 8x 2 ) + (3y 4y 2y) Group like terms together within each polynomial. 6x 2 4y + 5x 2 + y x 2 3y Use the vertical method. Simplify.
9 Add. Example 2D: Adding Polynomials Group like terms together.
10 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 Group like terms together. Combine like terms
11 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
12 Example 3A: Subtracting Polynomials Subtract. (x 3 + 4y) (2x 3 ) (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. Group like terms together.
13 Example 3B: Subtracting Polynomials Subtract. (7m 4 2m 2 ) (5m 4 5m 2 + 8) (7m 4 2m 2 ) + ( 5m 4 + 5m 2 8) (7m 4 2m 2 ) + ( 5m 4 + 5m 2 8) Rewrite subtraction as addition of the opposite. (7m 4 5m 4 ) + ( 2m 2 + 5m 2 ) 8 Group like terms together. 2m 4 + 3m 2 8
14 Example 3C: Subtracting Polynomials Subtract. ( 10x 2 3x + 7) (x 2 9) ( 10x 2 3x + 7) + ( x 2 + 9) ( 10x 2 3x + 7) + ( x 2 + 9) 10x 2 3x + 7 x 2 + 0x x 2 3x + 16 Rewrite subtraction as addition of the opposite. Use the vertical method. Write 0x as a placeholder.
15 Example 3D: Subtracting Polynomials Subtract. (9q 2 3q) (q 2 5) (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. Use the vertical method. Write 0 and 0q as placeholders.
16 Check It Out! Example 3 Subtract. (2x 2 3x 2 + 1) (x 2 + x + 1) (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. Use the vertical method. Write 0x as a placeholder.
17 Example 4: Application 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. (3x 2 + 7x 5) + (5x 2 4x + 11) 8x 2 + 3x + 6 Plot A. Plot B.
18 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 x x x x x x 3200 Eastern plant profit. Southern plant profit.
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