Let’s look more closely at the polynomial 8x3 9 2 x2 x 4. 1) The polynomial is written in descending powers of x since the powers of x decrease from left to right. Generally, we write polynomials in descending powers of the variable. 2) Recall that the term without a variable is called a constant. The constant is 9. The degree of a term equals the exponent on its variable. We will list each term, its coeffi cient, and its degree. Term Coefficient Degree 8x3 8 3 9 2 x2 9 2 2 x 1 1 4 4 0 (4 4x0) 3) The degree of the polynomial equals the highest degree of any nonzero term. The degree of 8x3 9 2 x2 x 4 is 3. Or, we say that this is a third-degree polynomial. EXAMPLE 1 Decide whether each expression is or is not a polynomial. If it is a polynomial, identify each term and the degree of each term. Then, fi nd the degree of the polynomial. a) 5p4 2.1p3 7p2 10 b) 3c2 1 5 c 2 8 c2 c) 7n6 Solution a) The expression 5p4 2.1p3 7p2 10 is a polynomial in p. Its terms have whole-number exponents and real coeffi cients. The term with the highest degree is 5p4, so the degree of the polynomial is 4. b) The expression 3c2 1 5 c 2 8 c2 is not a polynomial because one of its terms has a variable in the denominator. a 8 c2 8c2; the exponent 2 is not a whole number.b Term Degree 5p4 4 2.1p3 3 7p2 2 10 0 c) The expression 7n6 is a polynomial even though it has only one term. The degree of the term is 6, and that is the degree of the polynomial as well. In-Class Example 1 Decide whether each expression is or is not a polynomial. If it is a polynomial, identify each term and the degree of each term. Then, find the degree of the polynomial. a) 3a2 4 5 a 2 8 a2 b) 7k4 1.3k3 8k2 11 c) 9w5 Answer: a) not a polynomial b) polynomial Term Degree 7k4 4 1.3k3 3 8k2 2 11 0 degree: 4 c) polynomial Term Degree 9w5 5 degree: 5 YOU TRY 1 Decide whether each expression is or is not a polynomial. If it is a polynomial, identify each term and the degree of each term. Then, fi nd the degree of the polynomial. a) d 4 7d3 3 d b) k3 k2 3.8k 10 c) 2r 3r1/2 7 www.mhhe.com/messersmith SECTION 10.4 Adding and Subtracting Polynomials 791
messersmith_power_prealgebra_1e_ch4_7_10
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