COLLEGE ALGEBRA3.2 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 Far Left and Far Right Behavior

3.2 Far Left and Far Right Behavior

3.2 Far Left and Far Right BehaviorDetermine the far-right and far-left of the graphs.

3.2 Maximum and Minimum ValuesDetermine the far-right and far-left of the graphs.

3.2 Maximum and Minimum ValuesThe 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 ValuesThe 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 ValuesA 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 FunctionFactor to find the three real zeros of

Graph using software to verify.

3.2 Intermediate Value TheoremIf P is a polynomial function and P(a) ≠ P(b) for a < b, then P takes on every value between P(a) and P(b) in the interval [a,b].

We like to use the intermediate value theorem to help determine if there is a zero between two points. If one value comes up positive and one comes up negative it is safe to assume there is a zero in between.

Use the intermediate value theorem to verify that has a real zero between 1 and 2. Then use graphing software to verify.

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) = 0.

3. 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)• Using a graphing software, graph

The graph crosses at the following x = _______

However the graph intersects (touches) at the following x = _____

If c is a real number and the polynomial function P has (x – c) as a factor exactly k times, then the graph of P will…

1. intersect but not cross at the x-axis at (c, 0) provided k is an even positive integer.

2. cross the x-axis at (c, 0), provided that k is an odd positive integer.

3.2 Even and Odd Powers of (x – c)

Determine where the graph of crosses the x-axis and where the graph intersects, but does not cross the x-axis. Use graphing software to verify.

3.2 Procedure for Graphing Polynomial Functions

3.2 Procedure for Graphing Polynomial Functions

Sketch the graph of

3.3 Multiple Zeros of a Polynomial Function

Recall that if P is a polynomial function then the values of x for which P(x) is equal to 0 are called the zeros of P or the roots of the equation P(x) = 0. A zero of a polynomial function may be a multiple zero.

If a polynomial function P has (x – r) as a factor exactly k times, then r is a zero of multiplicity k of the polynomial function P.

_____ as a zero of multiplicity _____

_____ as a zero of multiplicity _____

_____ as a zero of multiplicity _____ -> referred to as a simple zero.

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 TheoremUse the Rational Zero Theorem to list all possible rational zeros of

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

According to the Upper – and Lower-Bound Theorem, what is the smallest positive integer that is an upper bound and the largest negative integer that is a lower bound of the real zeros of

3.3 Descartes’ Rule of SignsDescartes’ 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 SignsUse Descartes’ Rule of Signs to determine both the number of possible positive and the number of possible negative real numbers of each polynomial functions.

1.

2. 7

3.3 Zeros of a Polynomial Function

3.3 Zeros of a Polynomial Function

Find the zeros of

3.3 Applications

Glasses can be stacked to form a triangular pyramid. The total number of glasses in one of these pyramids is given by where k is the number of levels in the pyramid. If 220 glasses are used to form a triangular pyramid, how many levels are in the pyramid?

3.4 Fundamental Theorem of Algebra

If P is a polynomial function of degree n ≥ 1 with complex coefficients, then P has at least one complex zero.

This implies that P has a complex zero – say, . The Factor Theorem implies that

Where Q(x) is a polynomial of a degree one less than the degree of P. (Reduced polynomial)

3.4 Fundamental Theorem of Algebra

3.4 Fundamental Theorem of AlgebraFind all the zeros of each of the following polynomial functions, and write each function as a product of its leading coefficient and its linear functions.

1.

3.4 Conjugate Pair TheoremIn the previous example we notice that there were complex conjugates. This is not a coincidence.

Find all the zeros of given that is a zero.

3.4 Conjugate Pair TheoremSolve using a graphing software.

3.4 Conjugate Pair TheoremFind the polynomial function of degree 3 that has 1, 2, and -3 as zeros.

Homework13 – 47 Odd