Section 4.1 Polynomial Functions and Modeling Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives  Determine the behavior of the graph of a polynomial function using the leading-term test.  Factor polynomial functions and find their zeros and their multiplicities.  Use a graphing calculator to graph a polynomial function and find its real-number zeros.  Solve applied problems using polynomial models; fit linear, quadratic, power, cubic, and quartic polynomial functions to data.

A polynomial function P is given by where the coefficients a n, a n - 1, …, a 1, a 0 are real numbers and the exponents are whole numbers. Polynomial Function

Quadratic Function

Cubic Function

Examples of Polynomial Functions

Examples of Nonpolynomial Functions

Polynomial Functions The graph of a polynomial function is continuous and smooth. The domain of a polynomial function is the set of all real numbers.

The Leading-Term Test

Example Using the leading term-test, match each of the following functions with one of the graphs A  D, which follow. a)b) c)d)

Graphs a. b. c. d.

Solution CNegativeEven d)  x 6 APositiveOddc) x 5 BNegativeOdd b)  5x 3 DPositiveEven a) 3x 4 Graph Sign of Leading Coeff. Degree of Leading Term Leading Term

Graphs

Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f (c) = 0), then (c, 0) is an x-intercept of the graph of the function.

Example Find the zeros of To solve the equation f(x) = 0, we use the principle of zero products, solving x  1 = 0 and x + 2 = 0. The zeros of f(x) are 1 and  2. See graph on right.

Even and Odd Multiplicity If (x  c) k, k  1, is a factor of a polynomial function P(x) and (x  c) k + 1 is not a factor and:  k is odd, then the graph crosses the x-axis at (c, 0);  k is even, then the graph is tangent to the x-axis at (c, 0).

Example Find the zeros of f (x) = x 3 – 2x 2 – 9x Solution We factor by grouping. f (x) = x 3 – 2x 2 – 9x + 18 = x 2 (x – 2) – 9(x – 2). By the principle of zero products, the solutions of the equation f(x) = 0, are 2, –3, and 3.

Example Find the zeros of f (x) = x 4 + 8x 2 – 33. We factor as follows: f (x) = x 4 + 8x 2 – 33 = (x )(x 2 – 3). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products.

Example Find the zeros of f (x) = 0.2x 3 – 1.5x 2 – 0.3x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x-axis. We use the ZERO feature to find them. The zeros are approximately –1.164, 1.142, and –1010

Example The polynomial function can be used to estimate the number of milligrams of the pain relief medication ibuprofen in the bloodstream t hours after 400 mg of the medication has been taken. Find the number of milligrams in the bloodstream at t = 0, 0.5, 1, 1.5, and so on, up to 6 hr. Round the function values to the nearest tenth.

Example-continued Solution Using a calculator, we compute the function values.