COLLEGE ALGEBRA 3.2 Polynomial Functions of Higher Degree 3.3 Zeros of Polynomial Functions 3.4 Fundamental Theorem of Algebra

3.2 Polynomial Functions of Higher Degrees The following table describes everything we had talked about in Chapter 2. All polynomials have graphs that have smooth continuous curves. There are no breaks, holes or sharp turns…we will leave that for calculus

3.2 Far Left and Far Right Behavior

3.2 Maximum and Minimum Values

The graph below shows us two turning points in which the graph of the function changes from an increasing function to a decreasing function or vice versa.

3.2 Maximum and Minimum Values The minimum value of a function f is the smallest range value of f, it is called the absolute minimum. The maximum value of a function f is the largest range value of f, it is called the absolute maximum. We also have points, such as turning points which aren’t necessarily the absolute values, but are rather called relative max/min.

3.2 Maximum and Minimum Values A rectangular piece of cardboard measures 12 inches by 16 inches. An open box is formed by cutting squares that measure x inches by x inches from each of the corners of the cardboard and folding up the sides, as shown on page 275. Express the volume of the box as a function of x. Using graphing software, determine the x value that maximizes the volume.

3.2 Real Zeros of a Polynomial Function

3.2 Intermediate Value Theorem

3.2 Real Zeros, x- int., and Factors of a Polynomial Function If P is a polynomial function and c is a real number, then all of the following statements are equivalent in the following sense: If any one statement is true, then they all are true, and if any one statement is false, then they all are false. 1. (x – c) is a factor of P. 2. x = c is a real solution of P(x) = x = c is a real zero of P. 4. (c, 0) is an x-intercept of the graph of y = P(x)/

3.2 Real Zeros, x- int., and Factors of a Polynomial Function

3.2 Even and Odd Powers of (x – c)

3.2 Procedure for Graphing Polynomial Functions

3.3 Multiple Zeros of a Polynomial Function

Number of Zeros of a Polynomial Function A polynomial function P of degree n has at most n zeros, where each zero of multiplicity k is counted k times. Rational Zero Theorem

3.3 Rational Zero Theorem

3.3 Upper and Lower Bounds for Real Zeros A real number b is called an upper bound of the zeros of the polynomial function P if no zero is greater than b. A real number b is called a lower bound of the zeros of P if no zero is less than b.

3.3 Upper and Lower Bounds for Real Zeros

3.3 Descartes’ Rule of Signs Descartes’ Rule of Signs is another theorem that is often used to obtain information about the zeros of a polynomial function. The number of variations in sign of the coefficients refers to sign changes of the coefficients from positive to negative or negative to positive. **Must be written in descending order.

3.3 Descartes’ Rule of Signs

3.3 Zeros of a Polynomial Function

3.3 Applications

3.4 Fundamental Theorem of Algebra

3.4 Conjugate Pair Theorem

Find the polynomial function of degree 3 that has 1, 2, and -3 as zeros.

Homework 13 – 47 Odd