Chapter 5 Polynomials, Polynomial Functions, and Factoring

§ 5.1 Introduction to Polynomials and Polynomial Functions

Blitzer, Algebra for College Students, 6e – 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.

Blitzer, Algebra for College Students, 6e – Slide #4 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, Algebra for College Students, 6e – Slide #5 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.

Blitzer, Algebra for College Students, 6e – Slide #6 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, Algebra for College Students, 6e – Slide #7 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, Algebra for College Students, 6e – 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, Algebra for College Students, 6e – 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, Algebra for College Students, 6e – 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, Algebra for College Students, 6e – Slide #11 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, Algebra for College Students, 6e – Slide #12 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, Algebra for College Students, 6e – Slide #13 Section 5.1 Adding PolynomialsEXAMPLE SOLUTION Add: Remove parentheses Rearrange terms so that like terms are adjacent Combine like terms

Blitzer, Algebra for College Students, 6e – Slide #14 Section 5.1 Adding Polynomials YOUR TURN SOLUTION Add:

Blitzer, Algebra for College Students, 6e – Slide #15 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

Blitzer, Algebra for College Students, 6e – Slide #16 Section 5.1 Subtracting PolynomialsEXAMPLE SOLUTION Subtract

5.1 Assignment p.312 (2-24 even, even)