5.2 Polynomials

Introduction

Key results about polynomial functions

Using zeros and signs to understand polynomial behavior

The polyomial function $P(x) = 3 - 7x + 4x^2 - 2x^3 + 9x^5$ has degree $5$, its constant term is $3$, and its linear term is $-7x$.

Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zy. There you'll find a degree $4$ polynomial of the form $p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4$, where $a_0, \ldots, a_4$ are set up as sliders. In the questions that follow, you'll experiment with different values of $a_0, \ldots, a_4$ to investigate different possible behaviors in a degree $4$ polynomial. Note that we require $a_4 \ne 0$ in order to ensure $p$ is a degree $4$ polynomial.

What is the largest number of distinct points at which $p(x)$ can cross the $x$-axis?

Recall from the definition of a polynomial function what we mean by a zero of the polynomial. Give examples of values for $a_0, \ldots, a_4$ that lead to that largest number of zeros for $p(x)$.

A polynomial $p$ of degree $5$ with exactly $3$ real zeros, $4$ turning points, and such that $\lim_{x \to -\infty} p(x) = +\infty$ and $\lim_{x \to \infty} p(x) = -\infty$.

A polynomial $p$ of degree $4$ with exactly $4$ real zeros, $3$ turning points, and such that $\lim_{x \to -\infty} p(x) = +\infty$ and $\lim_{x \to \infty} p(x) = -\infty$.

A polynomial $p$ of degree $6$ with exactly $2$ real zeros, $3$ turning points, and such that $\lim_{x \to -\infty} p(x) = -\infty$ and $\lim_{x \to \infty} p(x) = -\infty$.

A polynomial $p$ of degree $5$ with exactly $5$ real zeros, $3$ turning points, and such that $\lim_{x \to -\infty} p(x) = +\infty$ and $\lim_{x \to \infty} p(x) = -\infty$.

Consider the polynomial function $p(x) = k(x-1)(x-a)(x-b)$. Suppose we know that $1 \lt a \lt b$ and that $k \lt 0$. Fully describe the graph of $p$ without the aid of a graphing utility.

What is the degree of $p$? How can you tell without fully expanding the factored form of the function?

What can you say about the sign of the factor $(x^2 + 10000)$?

What are the zeros of the polynomial $p$?

Construct a sign chart for $p$ by using the zeros you identified in (c) and then analyzing the sign of each factor of $p$.

Without using a graphing utility, construct an approximate graph of $p$ that has the zeros of $p$ carefully labeled on the $x$-axis.

Use a graphing utility to check your earlier work. What is challenging or misleading when using technology to graph $p$?