SECTION A.3 Polynomials A23 In Words A polynomial is a sum of monomials. Examples of Expressions that are Not Monomials (a) x3 1 2 is not a monomial, since the exponent of the variable x is 1 2 , and 1 2 is not a nonnegative integer. (b) x4 3− is not a monomial, since the exponent of the variable x is 3, − and 3− is not a nonnegative integer. EXAMPLE 2 Now Work problem 15 2 Recognize Polynomials Two monomials with the same variable raised to the same power are called like terms . For example, x2 4 and x5 4 − are like terms. In contrast, the monomials x2 3 and x2 5 are not like terms. We can add or subtract like terms using the Distributive Property. For example, x x x x x x x x 2 5 2 5 7 and 8 5 8 5 3 2 2 2 2 3 3 3 3 ( ) ( ) + = + = − = − = The sum or difference of two monomials having different degrees is called a binomial . The sum or difference of three monomials with three different degrees is called a trinomial . For example, • x 2 2 − is a binomial. • x x3 5 3 − + is a trinomial. • x x x 2 5 2 7 2 2 2 2 + + = + is a binomial. DEFINITION Polynomial A polynomial in one variable is an algebraic expression of the form a x a x a x a n n n n 1 1 1 0 + + + + − − (1) where a a a a , , . . . , , n n 1 1 0 − are constants,* called the coefficients of the polynomial, ≥n 0 is an integer, and x is a variable. If ≠ a 0, n it is called the leading coefficient , a xn n is called the leading term , and n is the degree of the polynomial. The monomials that make up a polynomial are called its terms . If all of the coefficients are 0, the polynomial is called the zero polynomial , which has no degree. Polynomials are usually written in standard form, beginning with the nonzero term of highest degree and continuing with terms in descending order according to degree. If a power of x is missing, it is because its coefficient is zero. Examples of Polynomials Polynomial Coefficients Degree − + − + x x x 8 4 6 2 3 2 − − 8, 4, 6, 2 3 − = + ⋅ − x x x 3 5 3 0 5 2 2 − 3, 0, 5 2 − + = ⋅ − + x x x x 8 2 1 2 8 2 2 − 1, 2, 8 2 + = + x x 5 2 5 2 1 5, 2 1 = ⋅ = ⋅ x 3 3 1 3 0 3 0 0 0 No degree EXAMPLE 3 *The notation a n is read as “ a sub n .” The number n is called a subscript and should not be confused with an exponent. We use subscripts to distinguish one constant from another when a large or undetermined number of constants are required.
RkJQdWJsaXNoZXIy NjM5ODQ=