Chapter 5 Polynomials, Polynomial Functions, and Factoring

§ 5.1 Introduction to Polynomials and Polynomial Functions

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.1 Polynomials A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents. Consider the polynomial: This polynomial contains four terms. It is customary to write the terms in order of descending powers of the variable. This is the standard form of a polynomial. Terms

Polynomials Not Polynomials A polynomial is a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in denominators. Polynomials

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.1 Polynomials The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of a term is (n +m) and the coefficient of the term is a. If there is exactly one term of greatest degree, it is called the leading term. It’ s coefficient is called the leading coefficient. Consider the polynomial: 3 is the leading coefficient. The degree is 4.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.1 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.

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.1 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) = = = = 0

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.1 Polynomials is an example of a polynomial function. In a polynomial function, the expression that defines the function is a polynomial. How do you evaluate a polynomial function? Use substitution just as you did to evaluate functions in Chapter 2.

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.1 PolynomialsEXAMPLE SOLUTION The polynomial function models the cumulative number of deaths from AIDS in the United States, f (x), x years after Use this function to solve the following problem. Find and interpret f (8). To find f (8), we replace each occurrence of x in the function’s formula with 8.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.1 Polynomials Thus, f (8) = 426,928. According to this model, this means that 8 years after 1990, in 1998, there had been 426,928 cumulative deaths from AIDS in the United States. CONTINUED Original function Replace each occurrence of x with 8 Evaluate exponents Multiply Add

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 4.1 Polynomials Check Point 2

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.1 Polynomials Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. By smooth, we mean that the graph contains only rounded corners with no sharp corners. By continuous, we mean that the graph has no breaks and can be drawn without lifting the pencil from the rectangular coordinate system.

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.1 Graphs of PolynomialsEXAMPLE The graph below does not represent a polynomial function. Although it has a couple of smooth, rounded corners, it also has a sharp corner and a break in the graph. Either one of these last two features disqualifies it from being a polynomial function. Smooth rounded curve Discontinuous break Sharp Corner

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.1 Adding PolynomialsEXAMPLE SOLUTION Add: Remove parentheses Rearrange terms so that like terms are adjacent Combine like terms Polynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 4.1 Adding Polynomials p 311 Check Point 6

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 4.1 Adding Polynomials Check Point 7

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.1 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 To subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 4.1 Subtracting Polynomials Check Point 8

DONE

Blitzer, Intermediate Algebra, 5e – Slide #20 Section 5.1 Graphs of Polynomials The Leading Coefficient Test As x increases or decreases without bound, the graph of a polynomial function eventually rises or falls. In particular, Odd-Degree Polynomials If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Even-Degree Polynomials If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right.

Blitzer, Intermediate Algebra, 5e – Slide #21 Section 5.1 PolynomialsEXAMPLE SOLUTION The common cold is caused by a rhinovirus. After x days of invasion by the viral particles, the number of particles in our bodies, f (x), in billions, can be modeled by the polynomial function Use the Leading Coefficient Test to determine the graph’s end behavior to the right. What does this mean about the number of viral particles in our bodies over time? Since the polynomial function has even degree and has a negative leading coefficient, the graph falls to the right (and the left). This means that the viral particles eventually decrease as the days increase.

Blitzer, Intermediate Algebra, 5e – Slide #22 Section 5.1 Polynomials The Degree of T If, the degree of is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree. Adding Polynomials Polynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms. Subtracting Polynomials To subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.