Adding and Subtracting Polynomials 6 4 Polynomials Warm

Adding and Subtracting Polynomials 6 -4 Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra Holt Algebra 1 1 Algebra 1

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

6 -4 Adding and Subtracting Polynomials Objective Add and subtract polynomials. Holt Algebra 1

6 -4 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. Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Example 1: Adding and Subtracting Monomials Add or subtract. A. 12 p 3 + 11 p 2 + 8 p 3 12 p 3 + 8 p 3 + 11 p 2 20 p 3 + 11 p 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. B. 5 x 2 – 6 – 3 x + 8 5 x 2 – 3 x + 8 – 6 5 x 2 – 3 x + 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. Holt Algebra 1

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

6 -4 Adding and Subtracting Polynomials 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. Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Check It Out! Example 1 Add or subtract. a. 2 s 2 + 3 s 2 + s 5 s 2 + s Identify like terms. Combine like terms. b. 4 z 4 – 8 + 16 z 4 + 2 4 z 4 + 16 z 4 – 8 + 2 20 z 4 – 6 Holt Algebra 1 Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

6 -4 Adding and Subtracting Polynomials Check It Out! Example 1 Add or subtract. c. 2 x 8 + 7 y 8 – x 8 – y 8 2 x 8 – x 8 + 7 y 8 – y 8 x 8 + 6 y 8 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. d. 9 b 3 c 2 + 5 b 3 c 2 – 13 b 3 c 2 b 3 c 2 Holt Algebra 1 Identify like terms. Combine like terms.

6 -4 Adding and Subtracting Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: 5 x 2 + 4 x + 1 + 2 x 2 + 5 x + 2 7 x 2 + 9 x + 3 In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms. (5 x 2 + 4 x + 1) + (2 x 2 + 5 x + 2) = (5 x 2 + 2 x 2) + (4 x + 5 x) + (1 + 2) = 7 x 2 + 9 x + 3 Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Example 2: Adding Polynomials Add. A. (4 m 2 + 5) + (m 2 – m + 6) Identify like terms. (4 m 2 + m 2) + (–m) +(5 + 6) Group like terms together. Combine like terms. 5 m 2 – m + 11 B. (10 xy + x) + (– 3 xy + y) Identify like terms. (10 xy – 3 xy) + x + y Group like terms together. Combine like terms. 7 xy + x + y Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Example 2 C: Adding Polynomials Add. (6 x 2 – 4 y) + (3 x 2 + 3 y – 8 x 2 – 2 y) Identify like terms. (6 x 2 – 4 y) + (– 5 x 2 + y) (6 x 2 – 5 x 2) + (– 4 y + y) x 2 – 3 y Holt Algebra 1 Combine like terms in the second polynomial. Combine like terms. Simplify.

6 -4 Adding and Subtracting Polynomials Example 2 D: Adding Polynomials Add. Identify like terms. Group like terms together. Combine like terms. Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Check It Out! Example 2 Add (5 a 3 + 3 a 2 – 6 a + 12 a 2) + (7 a 3 – 10 a) Identify like terms. (5 a 3 + 7 a 3) + (3 a 2 + 12 a 2) + (– 10 a – 6 a) Group like terms together. 12 a 3 + 15 a 2 – 16 a Holt Algebra 1 Combine like terms

6 -4 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: –(2 x 3 – 3 x + 7)= – 2 x 3 + 3 x – 7 Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Example 3 A: Subtracting Polynomials Subtract. (x 3 + 4 y) – (2 x 3) (x 3 + 4 y) + (– 2 x 3) Rewrite subtraction as addition of the opposite. Identify like terms. (x 3 – 2 x 3) + 4 y Group like terms together. –x 3 + 4 y Combine like terms. (x 3 + 4 y) + (– 2 x 3) Holt Algebra 1

6 -4 Adding and Subtracting Polynomials Example 3 B: Subtracting Polynomials Subtract. (7 m 4 – 2 m 2) – (5 m 4 – 5 m 2 + 8) (7 m 4 – 2 m 2) + (– 5 m 4 + 5 m 2 – 8) Rewrite subtraction as addition of the opposite. (7 m 4 – 2 m 2) + (– 5 m 4 + 5 m 2 – 8) Identify like terms. (7 m 4 – 5 m 4) + (– 2 m 2 + 5 m 2) – 8 Group like terms together. 2 m 4 + 3 m 2 – 8 Holt Algebra 1 Combine like terms.

6 -4 Adding and Subtracting Polynomials Example 3 C: Subtracting Polynomials Subtract. (– 10 x 2 – 3 x + 7) – (x 2 – 9) (– 10 x 2 – 3 x + 7) + (–x 2 + 9) – 10 x 2 – 3 x + 7 –x 2 + 0 x + 9 – 11 x 2 – 3 x + 16 Holt Algebra 1 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0 x as a placeholder. Combine like terms.

6 -4 Adding and Subtracting Polynomials Example 3 D: Subtracting Polynomials Subtract. (9 q 2 – 3 q) – (q 2 – 5) (9 q 2 – 3 q) + (–q 2 + 5) 9 q 2 – 3 q + 0 + − q 2 – 0 q + 5 8 q 2 – 3 q + 5 Holt Algebra 1 Rewrite subtraction as addition of the opposite. Identify like terms. Use the vertical method. Write 0 and 0 q as placeholders. Combine like terms.

6 -4 Adding and Subtracting Polynomials Check It Out! Example 3 Subtract. (2 x 2 – 3 x 2 + 1) – (x 2 + x + 1) (2 x 2 – 3 x 2 + 1) + (–x 2 – x – 1) Rewrite subtraction as addition of the opposite. (2 x 2 – 3 x 2 + 1) + (–x 2 – x – 1) Identify like terms. –x 2 + 0 x + 1 + –x 2 – x – 1 – 2 x 2 – x Holt Algebra 1 Use the vertical method. Write 0 x as a placeholder. Combine like terms.

6 -4 Adding and Subtracting Polynomials 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 3 x 2 + 7 x – 5 and the area of plot B can be represented by 5 x 2 – 4 x + 11. Write a polynomial that represents the total area of both plots of land. (3 x 2 + 7 x – 5) + (5 x 2 – 4 x + 11) 8 x 2 + 3 x + 6 Holt Algebra 1 Plot A. Plot B. Combine like terms.

6 -4 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. 03 x 2 + 25 x – 1500 + – 0. 02 x 2 + 21 x – 1700 – 0. 05 x 2 + 46 x – 3200 Holt Algebra 1 Eastern plant profit. Southern plant profit. Combine like terms.

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

6 -4 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 the first wall is modeled by 4 x 2 + 12 x + 9, and the area of the second wall is modeled by 36 x 2 – 12 x + 1. Write a polynomial that represents the total area of the two walls. 40 x 2 + 10 Holt Algebra 1 of