1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 Section 5.4 Polynomials in Several Variables Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3. 3 A polynomial containing two or more variables is called a polynomial in several variables. An example of a polynomial in two variables is: Polynomials in Several Variables

4. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 1.Substitute the given value for each variable. 2.Perform the resulting computation using the order of operations. Evaluating a Polynomial in Several Variables

5. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 Evaluate 5x 3 y + 6xy − 2x 2 y for x = 3 and y = −1. 1.Substitute the given value for each variable. 2.Perform the resulting computation using the order of operations. Evaluating a Polynomial in Several VariablesEXAMPLE

6. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 Evaluate 5x 3 y + 6xy − 2x 2 y for x = 3 and y = −1. 1.Substitute the given value for each variable. 2.Perform the resulting computation using the order of operations. Evaluating a Polynomial in Several VariablesEXAMPLE

7. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 Objective #1: Example

8. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 Objective #1: Example

9. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9

10. 10 Polynomials In general, a polynomial in two variables, x and y, contains the sum of one or more monomials in the form The constant, a, is the coefficient. The exponents, n and m, represent whole numbers. The degree of the monomial is n + m.

11. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 PolynomialsEXAMPLE SOLUTION Determine the coefficient of each term, the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient of the polynomial. TermCoefficientDegree (Sum of Exponents on the Variables) 124 + 1 = 5 3 + 7 = 10 2 + 0 = 2 440 + 0 = 0

12. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 PolynomialsCONTINUED The degree of the polynomial is the greatest degree of all its terms, which is 10. The leading term is the term of the greatest degree, which is. Its coefficient, − 5, is the leading coefficient.

13. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Objective #2: Example

14. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Objective #2: Example

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16. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Polynomials in several variables are added by combining like terms. Polynomials in several variables are subtracted by adding the first polynomial and the opposite of the second polynomial.  Like terms are terms containing exactly the same variables to the same powers. Adding and Subtracting Polynomials in Several Variables

17. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Subtracting PolynomialsEXAMPLE SOLUTION Subtract Change subtraction to addition and change the sign of every term of the polynomial in parentheses. Rearrange terms Combine like terms

18. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Objective #3: Example

19. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 Objective #3: Example

20. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20 Objective #3: Example

21. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21 Objective #3: Example

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23. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23 Multiplying Polynomials in Several Variables The product of monomials forms the basis of polynomial multiplication. As with monomials in one variable, multiplication can be done mentally by multiplying coefficients and adding exponents on variables with the same base.

24. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Multiply coefficients and add exponents on variables with the same base. Regroup. Multiply the coefficients and add the exponents. Multiplying Polynomials in Several VariablesEXAMPLE

25. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Multiply each term of the polynomial by the monomial. Use the distributive property. Multiply the coefficients and add the exponents. Multiplying Polynomials in Several VariablesEXAMPLE

26. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26 Objective #4: Example

27. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27 Objective #4: Example

28. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 Objective #4: Example

29. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 29 Objective #4: Example

30. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Objective #4: Example

31. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Objective #4: Example

32. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Objective #4: Example

33. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Objective #4: Example