Adding and Subtracting Polynomials 6-4 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up Simplify each expression by combining like terms. 1. 4x + 2x 2. 3y + 7y 3. 8p – 5p 4. 5n + 6n2 Simplify each expression. 5. 3(x + 4) 6. –2(t + 3) 7. –1(x2 – 4x – 6) 6x 10y 3p not like terms 3x + 12 –2t – 6 –x2 + 4x + 6

Objective Add and subtract polynomials.

Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

Example 1: Adding and Subtracting Monomials Add or subtract. A. 12p3 + 11p2 + 8p3 12p3 + 11p2 + 8p3 Identify like terms. Rearrange terms so that like terms are together. 12p3 + 8p3 + 11p2 20p3 + 11p2 Combine like terms. B. 5x2 – 6 – 3x + 8 Identify like terms. 5x2 – 6 – 3x + 8 Rearrange terms so that like terms are together. 5x2 – 3x + 8 – 6 5x2 – 3x + 2 Combine like terms.

Example 1: Adding and Subtracting Monomials Add or subtract. C. t2 + 2s2 – 4t2 – s2 t2 + 2s2 – 4t2 – s2 Identify like terms. Rearrange terms so that like terms are together. t2 – 4t2 + 2s2 – s2 –3t2 + s2 Combine like terms. D. 10m2n + 4m2n – 8m2n 10m2n + 4m2n – 8m2n Identify like terms. 6m2n Combine like terms.

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!

Check It Out! Example 1 Add or subtract. a. 2s2 + 3s2 + s 2s2 + 3s2 + s Identify like terms. 5s2 + s Combine like terms. b. 4z4 – 8 + 16z4 + 2 4z4 – 8 + 16z4 + 2 Identify like terms. Rearrange terms so that like terms are together. 4z4 + 16z4 – 8 + 2 20z4 – 6 Combine like terms.

Check It Out! Example 1 Add or subtract. c. 2x8 + 7y8 – x8 – y8 Identify like terms. 2x8 + 7y8 – x8 – y8 Rearrange terms so that like terms are together. 2x8 – x8 + 7y8 – y8 x8 + 6y8 Combine like terms. d. 9b3c2 + 5b3c2 – 13b3c2 9b3c2 + 5b3c2 – 13b3c2 Identify like terms. b3c2 Combine like terms.

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. 5x2 + 4x + 1 + 2x2 + 5x + 2 7x2 + 9x + 3 (5x2 + 4x + 1) + (2x2 + 5x + 2) = (5x2 + 2x2) + (4x + 5x) + (1 + 2) = 7x2 + 9x + 3

Example 2: Adding Polynomials A. (4m2 + 5) + (m2 – m + 6) (4m2 + 5) + (m2 – m + 6) Identify like terms. Group like terms together. (4m2 + m2) + (–m) +(5 + 6) 5m2 – m + 11 Combine like terms. B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) Identify like terms. Group like terms together. (10xy – 3xy) + x + y 7xy + x + y Combine like terms.

Example 2C: Adding Polynomials (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) Identify like terms. Combine like terms in the second polynomial. (6x2 – 4y) + (–5x2 + y) (6x2 –5x2) + (–4y + y) Combine like terms. x2 – 3y Simplify.

Example 2D: Adding Polynomials Identify like terms. Group like terms together. Combine like terms.

Check It Out! Example 2 Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a). (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a) Identify like terms. (5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a) Group like terms together. 12a3 + 15a2 – 16a Combine like terms.

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: –(2x3 – 3x + 7)= –2x3 + 3x – 7

Example 3A: Subtracting Polynomials (x3 + 4y) – (2x3) Rewrite subtraction as addition of the opposite. (x3 + 4y) + (–2x3) (x3 + 4y) + (–2x3) Identify like terms. (x3 – 2x3) + 4y Group like terms together. –x3 + 4y Combine like terms.

Example 3B: Subtracting Polynomials (7m4 – 2m2) – (5m4 – 5m2 + 8) (7m4 – 2m2) + (–5m4 + 5m2 – 8) Rewrite subtraction as addition of the opposite. (7m4 – 2m2) + (–5m4 + 5m2 – 8) Identify like terms. Group like terms together. (7m4 – 5m4) + (–2m2 + 5m2) – 8 2m4 + 3m2 – 8 Combine like terms.

Example 3C: Subtracting Polynomials (–10x2 – 3x + 7) – (x2 – 9) (–10x2 – 3x + 7) + (–x2 + 9) Rewrite subtraction as addition of the opposite. (–10x2 – 3x + 7) + (–x2 + 9) Identify like terms. –10x2 – 3x + 7 –x2 + 0x + 9 Use the vertical method. Write 0x as a placeholder. –11x2 – 3x + 16 Combine like terms.

Example 3D: Subtracting Polynomials (9q2 – 3q) – (q2 – 5) Rewrite subtraction as addition of the opposite. (9q2 – 3q) + (–q2 + 5) (9q2 – 3q) + (–q2 + 5) Identify like terms. Use the vertical method. 9q2 – 3q + 0 + − q2 – 0q + 5 Write 0 and 0q as placeholders. 8q2 – 3q + 5 Combine like terms.

Check It Out! Example 3 Subtract. (2x2 – 3x2 + 1) – (x2 + x + 1) Rewrite subtraction as addition of the opposite. (2x2 – 3x2 + 1) + (–x2 – x – 1) (2x2 – 3x2 + 1) + (–x2 – x – 1) Identify like terms. Use the vertical method. –x2 + 0x + 1 + –x2 – x – 1 Write 0x as a placeholder. –2x2 – x Combine like terms.

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 3x2 + 7x – 5 and the area of plot B can be represented by 5x2 – 4x + 11. Write a polynomial that represents the total area of both plots of land. (3x2 + 7x – 5) Plot A. + (5x2 – 4x + 11) Plot B. 8x2 + 3x + 6 Combine like terms.

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. –0.03x2 + 25x – 1500 Eastern plant profit. + –0.02x2 + 21x – 1700 Southern plant profit. –0.05x2 + 46x – 3200 Combine like terms.

Lesson Quiz: Part I Add or subtract. 1. 7m2 + 3m + 4m2 2. (r2 + s2) – (5r2 + 4s2) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d2 – 8) + (6d2 – 2d +1) 11m2 + 3m (–4r2 – 3s2) 18pq – 2p 20d2 – 2d – 7 5. (2.5ab + 14b) – (–1.5ab + 4b) 4ab + 10b

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 4x2 + 12x + 9, and the area of the second wall is modeled by 36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls. 40x2 + 10