Section 1: Polynomial Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 6.1 Polynomial Functions 549 Version: Fall 2007 6.1 Polynomial Functions We’ve seen in previous sections that a monomial is the product of a number and one or more variable factors, each raised to a positive integral power, as in 3 x 2 or 4 x 3 y 4 . We’ve also seen that a binomial is the sum or difference of two monomial terms, as in 3 x +5 , x 2 +4 , or 3 xy 2 2 x 2 y . We’ve also seen that a trinomial is the sum or difference of three monomial terms, as in x 2 2 x 3 or x 2 4 xy + 5 y 2 . The root word “poly” means “many,” as in polygon (many sides) or polyglot (speak- ing many languages—multilingual). In algebra, the word polynomial means “many terms,” where the phrase “many terms” can be construed to mean anywhere from one to an arbitrary, but finite, number of terms. Consequently, a monomial could be considered a polynomial, as could binomials and trinomials. In our work, we will concentrate for the most part on polynomials of a single variable. What follows is a more formal definition of a polynomial in a single variable x . Definition 1. The function p , defined by p ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n (2) is called a polynomial in x . There are several important points to be made about this definition. 1. The polynomial in our definition is arranged in ascending powers of x . We could just as easily arrange our polynomial in descending powers of x , as in p ( x ) = a n x n + · · · + a 2 x 2 + a 1 x + a 0 . 2. The numbers a 0 , a 1 , a 2 , . . . , a n are called the coefficients of the polynomial p . If all of the coefficients are integers, then we say that “ p is a polynomial with integer coefficients.” If all of the coefficients are rational numbers, then we say that “ p is a polynomial with rational coefficients.” If all of the coefficients are real numbers, then we say that “ p is a polynomial with real coefficients.” 3. The degree of the polynomial p is n , the highest power of x . 4. The leading term of the polynomial p is the term with the highest power of x . In the case of equation (2) , the leading term is a n x n . Let’s look at an example. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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550 Chapter 6 Polynomial Functions Version: Fall 2007 l⚏ Example 3. Consider the polynomial p ( x ) = 3 4 x 2 + 5 x 3 6 x. (4) Find the degree, the leading term, and make a statement about the coefficients of p . First, put the polynomial terms in order. Whether you use ascending or descending powers of x makes no difference. Choose one or the other. In descending powers of x , p ( x ) = 5 x 3 4 x 2 6 x + 3 , (5) but in ascending powers of x , p ( x ) = 3 6 x 4 x 2 + 5 x 3 . (6) In either case, equation (5) or equation (6) , the degree of the polynomial is 3 . Also, in either case, the leading term 2 of the polynomial is 5 x 3 . Because all coefficients of this polynomial are integers, we say that “ p is a polynomial with integer coefficients.” However, all the coefficients are also rational numbers, so we could say that p is a polynomial with rational coefficients. For that matter, all of the coefficients of
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