Write equations for several polynomial functions of odd...

Name: Problem 78, Chapter 3: Algebra and Trigonometry
Uploaded: 2020-06-18T13:51:13-08:00
Duration: 2 min 38 s
Channel: Derrick Hanson
Description: Write equations for several polynomial functions of odd degree and graph each function. Is it possible for the graph to have no real zeros? Explain. Try doing the same thing for polynomial functions of even degree. Now is it possible to have no real zeros?

Write equations for several polynomial functions of odd degree and graph each function. Is it possible for the graph to have no real zeros? Explain. Try doing the same thing for polynomial functions of even degree. Now is it possible to have no real zeros?

Key Concepts

Even Degree Polynomials

Even degree polynomials have exponents that are even numbers and typically display the same behavior at both ends of the graph (i.e., both tails either point up or down). This means that an even degree polynomial can possibly have no real zeros if the graph does not intersect the x-axis, such as when the entire graph lies above or below it.

Real Zeros

Real zeros of a polynomial function are the x-values for which the function evaluates to zero, meaning the graph of the function crosses or touches the x-axis at these points. Whether or not real zeros exist, and how many there may be, depends on the degree of the polynomial, its coefficients, and the nature of its factors.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable in the expression. The degree is a crucial factor in determining the shape of the graph of the function, its end behavior, and the maximum number of roots it can possess.

Polynomial Functions

A polynomial function is an expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions can be used to model a wide range of phenomena and are particularly useful because their behavior is well understood in terms of degree and coefficients.

Odd Degree Polynomials

Odd degree polynomials have exponents that are odd numbers and exhibit end behavior where the function tends to infinity in one direction and negative infinity in the opposite direction. This contrasting end behavior ensures that there is at least one real zero, as the function must cross the x-axis due to the Intermediate Value Theorem.

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