Adding and subtracting polynomials involves combining like terms. To add or subtract polynomials, identify like terms and combine them. For example, to add 12p^3 + 11p^2 + 8p^3, the like terms 12p^3 and 8p^3 are combined to get 20p^3, and 11p^2 is left unchanged. Polynomials can be added or subtracted either vertically by aligning like terms or horizontally by regrouping terms. The process is similar to adding and subtracting numbers.
2. Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.
3. Add or Subtract.. Example 1: Adding and Subtracting Monomials A. 12 p 3 + 11 p 2 + 8 p 3 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 – 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.
4. Add or Subtract.. Adding and Subtracting Monomials C. t 2 + 2 s 2 – 4 t 2 – s 2 t 2 – 4 t 2 + 2s 2 – s 2 t 2 + 2 s 2 – 4 t 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 10 m 2 n + 4 m 2 n – 8 m 2 n 6 m 2 n Identify like terms. Combine like terms.
5. Polynomials can be added and subtracted 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. ( 5 x 2 + 4 x + 1 ) – ( 2 x 2 + 5 x + 2 ) = 3 x 2 – x – 1 Example: Adding and Subtracting Polynomials 5 x 2 + 4 x + 1 + 2 x 2 + 5 x + 2 7 x 2 + 9 x + 3
6. Example : Adding and Subtracting Polynomials A. (4 m 2 + 5) + ( m 2 – m + 6) 4 m 2 + 5 + m 2 – m + 6 4 m 2 + m 2 – m + 5 + 6 5 m 2 – m + 11 Polynomials can be added and subtracted by combining and adding as monomials. B. (7 m 4 – 2 m 2 ) – (5 m 4 – 5 m 2 + 8)
9. 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. Example 4: Application (3 x 2 + 7 x – 5) (5 x 2 – 4 x + 11) Plot A. Plot B. Combine like terms. + 8 x 2 + 3 x + 6
10. Example 5 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 Eastern plant profit. – 0.02 x 2 + 21 x – 1700 Southern plant profit. Combine like terms. + – 0.05 x 2 + 46 x – 3200
