1. Section Concepts 2.1 Addition and Subtraction of Polynomials Slide 1 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Introduction to Polynomials 2.Addition of Polynomials 3.Subtraction of Polynomials

2. Section Concepts 2.1 Addition and Subtraction of Polynomials Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Any Homework Questions?

3. Section 2.1 Addition and Subtraction of Polynomials 1.Introduction to Polynomials Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. One commonly used algebraic expression is called a polynomial. A polynomial in one variable, x, is defined as a single term or a sum of terms of the form where a is a real number and the exponent, n, is a nonnegative integer. For each term, a is called the coefficient, and n is called the degree of the term.

4. Section 2.1 Addition and Subtraction of Polynomials 1.Introduction to Polynomials Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A polynomial which contains one term is categorized as a monomial. A polynomial which contains two terms is categorized as a binomial. A polynomial which contains three terms is categorized as a trinomial. An expression which contains four or more terms is categorized a polynomial.

5. Section 2.1 Addition and Subtraction of Polynomials 1.Introduction to Polynomials Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The degree of a polynomial is the highest power of all of its terms. Thus, when written in descending order, the leading term determines the degree of the polynomial.

6. Section 2.1 Addition and Subtraction of Polynomials 1.Introduction to Polynomials Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

7. Example 1Identifying the Parts of a Polynomial Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Given: a. List the terms of the polynomial, and state the coefficient and degree of each term. b. Write the polynomial in descending order. c. State the degree of the polynomial and the leading coefficient.

8. Example Solution: 1Identifying the Parts of a Polynomial Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. term: term: coefficient: degree: 4.5a 1.6 b. c. The degree of the polynomial is The leading coefficient is Write the polynomial in descending order.

9. Section 2.1 Addition and Subtraction of Polynomials 1.Introduction to Polynomials Slide 9 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Polynomials may have more than one variable. In such a case, the degree of a term is the sum of the exponents of the variables contained in the term. The following polynomial has a degree of 11 because the highest degree of its terms is 11.

10. Section 2.1 Addition and Subtraction of Polynomials 2.Addition of Polynomials Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that two terms are like terms if they each have the same variables, and the corresponding variables are raised to the same powers.

11. Section 2.1 Addition and Subtraction of Polynomials 2.Addition of Polynomials Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that the distributive property is used to add or subtract like terms. For example, We can shorten the distributive process by adding coefficients of like terms.

12. Example 2Adding Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Add the polynomials. a. b.

13. Example 3Adding Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Add the polynomials. a. b.

14. TIP: Slide 14 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Polynomials can also be added by combining like terms in columns. Place holders such as 0 and 0c may be used to help line up like terms. Add the polynomials ___________________

15. Example 4Adding Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Add the polynomials. a. b.

16. Section 2.1 Addition and Subtraction of Polynomials 3.Subtraction of Polynomials Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtraction of two polynomials requires us to find the opposite of the polynomial being subtracted. To find the opposite of a polynomial, take the opposite of each term. This is equivalent to multiplying the polynomial by

17. Example Solution: 5Finding the Opposite of a Polynomial Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Expression Opposite Simplified Form a.b. c.

18. TIP: Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notice that the sign of each term is changed when finding the opposite of a polynomial.

19. DEFINITIONSubtraction of Polynomials Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. If A and B are polynomials, then

20. Example 6Subtracting Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtract the polynomials. a. b.

21. TIP: Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two polynomials can also be subtracted in columns by adding the opposite of the second polynomial to the first polynomial. Place holders (shown in red) may be used to help line up like terms. The difference of the polynomials is

22. Example 7Subtracting Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtract the polynomials a. b.Find the difference ofand

23. Example 8Subtracting Polynomials Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Perform the indicated operations a.

24. Example 2.5 Addition and Subtraction of Polynomials Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Introduction to Polynomials 2.Addition of Polynomials 3.Subtraction of Polynomials

25. Example You Try: Add the polynomials. Subtract the polynomials

26. Example You Try: Perform the indicated operation3.