5.2 Polynomials (continued)

Multiplicity of polynomial zeros

Summary

A polynomial $f$ of degree $10$ whose zeros are $x = -12$ (multiplicity $3$), $x = -9$ (multiplicity $2$), $x = 4$ (multiplicity $4$), and $x = 10$ (multiplicity $1$), and $f$ satisfies $f(0) = 21$. What can you say about the values of $\lim_{x \to -\infty} f(x)$ and $\lim_{x \to \infty} f(x)$?

A polynomial $p$ of degree $9$ that satisfies $p(0) = -2$ and has the graph shown in the following figure. Assume that all of the zeros of $p$ are shown in the figure.

A polynomial $q$ of degree $8$ with $3$ distinct real zeros (possibly of different multiplicities) such that $q$ has the sign chart in the figure below and satisfies $q(0) = -10$.

A polynomial $q$ of degree $9$ with $3$ distinct real zeros (possibly of different multiplicities) such that $q$ satisfies the sign chart in part (c) and satisfies $q(0) = -10$.

A polynomial $p$ of degree $11$ that satisfies $p(0) = -2$ and $p$ has the graph shown in part (b). Assume that all of the zeros of $p$ are shown in the figure.

Consider the polynomial function given by

.

• What is the degree of $p$? - What are the real zeros of $p$? State them with multiplicity. - Construct a carefully labeled sign chart for $p(x)$. - Plot the function $p$ in Desmos. Are the zeros obvious from the graph? How do you have to adjust the window in order to tell? Even in an adjusted window, can you tell them exactly from the graph? - Now consider the related but different polynomial

. What is the degree of $q$? What are the zeros of $q$? What is obvious from its graph and what is not?

Consider the (non-polynomial) function $r(x) = e^{-x^2}(x^2+1)(x-2)(x-3)$.

• What are the zeros of $r(x)$? (Hint: is $e^{\Box}$ ever equal to zero?) - Construct a sign chart for $r(x)$. - Plot $r(x)$ in Desmos. Is the sign and overall behavior of $r$ obvious from the plot? Why or why not? - From the graph, what appears to be the value of $\lim_{x \to \infty} r(x)$? Why is this surprising in light of the behavior of $f(x)=(x^2+1)(x-2)(x-3)$ as $x \to \infty$?

In each following question, find a formula for a polynomial with certain properties, generate a plot that demonstrates you’ve found a function with the given specifications, and write several sentences to explain your thinking.

• A quadratic function $q$ has zeros at $x = −7$ and $x = 11$ and its $y$-value at its vertex is $42$. - A polynomial $r$ of degree $4$ has zeros at $x = −3$ and $x = 5$, both of multiplicity $2$, and the function has a $y$-intercept at the point $(0, 28)$. - A polynomial $f$ has degree $11$ and the following zeros: zeros of multiplicity $1$ at $x = −3$ and $x = 5$, zeros of multiplicity $2$ at $x = −2$ and $x = 3$, and a zero of multiplicity $3$ at $x = 1$. In addition, $\lim_{x \to \infty} f(x) = -\infty$. - A polynomial $g$ has its graph given in Figure below. Determine a possible formula for $g(x)$ where the polynomial you find has the lowest possible degree to match the graph. What is the degree of the function you find? A polynomial function $g$.

Like we have worked to understand families of functions that involve parameters such as $p(t) = a\cos(k(t-b)) + c$ and $F(t) = a + be^{-kt}$, we are often interested in polynomials that involve one or more parameters and understanding how those parameters affect the function's behavior.

For example, let $a \gt 0$ be a positive constant, and consider $p(x) = x^3 - a^2x$.

• What is the degree of $p$? - What is the long-term behavior of $p$? State your responses using limit notation. - In terms of the constant $a$, what are the zeros of $p$? - Construct a carefully labeled sign chart for $p$. - How does changing the value of $a$ affect the graph of $p$?