1. Introduction Polynomials, or expressions that contain variables, numbers, or combinations of variables and numbers, can be added and subtracted like real numbers. Adding and subtracting polynomials is a way to simplify expressions and find a different, but equivalent, way to represent a sum or difference. 1 2.1.2: Adding and Subtracting Polynomials

2. Key Concepts Like terms are terms that contain the same variables raised to the same power. For example, the terms 2a 3 and –9a 3 are like terms, since each contains the variable a raised to the third power. To add two polynomials, combine like terms by adding the coefficients of the terms. When two polynomials are added, the result is another polynomial. Since the sum of two polynomials is a polynomial, the group of polynomials is closed under the operation of addition. A system is closed, or shows closure, under an operation if the result of the operation is within the system. 2 2.1.2: Adding and Subtracting Polynomials

3. Key Concepts, continued For example, integers are closed under the operation of multiplication because the product of two integers is always an integer. However, integers are not closed under division, because in some cases the result is not an integer—for example, the result of 3 ÷ 2 is 1.5. To subtract two polynomials, first rewrite the subtraction using addition. For example, a + 1 – (a – 10) = a + 1 + (–a + 10). After rewriting using addition, combine like terms. 3 2.1.2: Adding and Subtracting Polynomials

4. Key Concepts, continued Subtraction of polynomials can be rewritten as addition, and polynomials are closed under the operation of addition; therefore, polynomials are also closed under the operation of subtraction. 4 2.1.2: Adding and Subtracting Polynomials

5. Guided Practice Example 2 Simplify (6x 4 – x 3 – 3x 2 + 20) + (10x 3 – 4x 2 + 9). 5 2.1.2: Adding and Subtracting Polynomials

6. Guided Practice: Example 2, continued 1.Rewrite any subtraction using addition. Subtraction can be rewritten as adding a negative. (6x 4 – x 3 – 3x 2 + 20) + (10x 3 – 4x 2 + 9) = [6x 4 + (–x 3 ) + (–3x 2 ) + 20] + [10x 3 + (–4x 2 ) + 9] 6 2.1.2: Adding and Subtracting Polynomials

7. Guided Practice: Example 2, continued 2.Rewrite the sum so that any like terms are together. Be sure to keep any negatives with the terms. [6x 4 + (–x 3 ) + (–3x 2 ) + 20] + [10x 3 + (–4x 2 ) + 9] = 6x 4 + (–x 3 ) + 10x 3 + (–3x 2 ) + (–4x 2 ) + 20 + 9 7 2.1.2: Adding and Subtracting Polynomials

8. Guided Practice: Example 2, continued 3.Find the sum of any constants. The previous expression contains two constants: 20 and 9. 6x 4 + (–x 3 ) + 10x 3 + (–3x 2 ) + (–4x 2 ) + 20 + 9 = 6x 4 + (–x 3 ) + 10x 3 + (–3x 2 ) + (–4x 2 ) + 29 8 2.1.2: Adding and Subtracting Polynomials

9. Guided Practice: Example 2, continued 4.Find the sum of any terms with the same variable raised to the same power. The previous expression contains the following like terms: (–x 3 ) and 10x 3 ; (–3x 2 ) and (–4x 2 ). 9 2.1.2: Adding and Subtracting Polynomials

10. Guided Practice: Example 2, continued Add the coefficients of any like terms, being sure to keep any negatives with the coefficients. 6x 4 + (–x 3 ) + 10x 3 + (–3x 2 ) + (–4x 2 ) + 29 = 6x 4 + (–1 + 10)x 3 + [–3 + (–4)]x 2 + 29 = 6x 4 + 9x 3 – 7x 2 + 29 (6x 4 – x 3 – 3x 2 + 20) + (10x 3 – 4x 2 + 9) is equivalent to 6x 4 + 9x 3 – 7x 2 + 29. 10 2.1.2: Adding and Subtracting Polynomials ✔

11. Guided Practice Example 3 Simplify (–x 6 + 7x 2 + 11) – (12x 6 + 4x 5 – 2x + 1). 11 2.1.2: Adding and Subtracting Polynomials

12. Guided Practice: Example 3, continued 1.Rewrite the difference as a sum. A difference can be written as the sum of a negative quantity. Distribute the negative in the second polynomial. (–x 6 + 7x 2 + 11) – (12x 6 + 4x 5 – 2x + 1) = (–x 6 + 7x 2 + 11) + [–(12x 6 + 4x 5 – 2x + 1)] = (–x 6 + 7x 2 + 11) + (–12x 6 ) + (–4x 5 ) + 2x + (–1) 12 2.1.2: Adding and Subtracting Polynomials

13. Guided Practice: Example 3, continued 2.Rewrite the sum so that any like terms are together. Be sure to keep any negatives with the coefficients. (–x 6 + 7x 2 + 11) + (–12x 6 ) + (–4x 5 ) + 2x + (–1) = –x 6 + (–12x 6 ) + (–4x 5 ) + 7x 2 + 2x + 11 + (–1) 13 2.1.2: Adding and Subtracting Polynomials

14. Guided Practice: Example 3, continued 3.Find the sum of any constants. The previous expression contains two constants: 11 and (–1). –x 6 + (–12x 6 ) + (–4x 5 ) + 7x 2 + 2x + 11 + (–1) = –x 6 + (–12x 6 ) + (–4x 5 ) + 7x 2 + 2x + 10 14 2.1.2: Adding and Subtracting Polynomials

15. Guided Practice: Example 3, continued 4.Find the sum of any terms with the same variable raised to the same power. The previous expression contains the following like terms: –x 6 and (–12x 6 ). –x 6 + (–12x 6 ) + (–4x 5 ) + 7x 2 + 2x + 10 = [(–1) + (–12)]x 6 + (–4x 5 ) + 7x 2 + 2x + 10 = –13x 6 + (–4x 5 ) + 7x 2 + 2x + 10 = –13x 6 – 4x 5 + 7x 2 + 2x + 10 (–x 6 + 7x 2 + 11) – (12x 6 + 4x 5 – 2x + 1) is equivalent to –13x 6 – 4x 5 + 7x 2 + 2x + 10. 15 2.1.2: Adding and Subtracting Polynomials ✔