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!

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 + 1) + (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 C. (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) (6x2 – 4y) + (3x2 + 3y – 8x2 – 2y) Identify like terms. Group like terms together within each polynomial. (6x2 + 3x2 – 8x2) + (3y – 4y – 2y) 6x2 – 4y + –5x2 + y Use the vertical method. Combine like terms. x2 – 3y Simplify.

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.

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.

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