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

Add and subtract polynomials. Objective

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

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.

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.

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 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.

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.

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

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.

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

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

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.

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

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.

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.

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.

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.

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.

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. +

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

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

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