In this exercise, use a calculator to find the coordinates of the vertex of the graph. Express: coordinates to the nearest hundredth. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$

Find a polynomial function $P(x)$ having leading coefficient one, least possible degree, real coefficients, and the given zeros.

$5$ and $-4$

Solve the given equation. For equations with real solutions, support your answers graphically.

$$(3 x-1)^2=12$$

Answer the following given problems.

Suppose that a polynomial function $P$ is defined in such a way that $P(2)=-4$ and $P(2.5)=2$. What conclusion does the intermediate value theorem allow you to make?

Solve the given equation. For equations with real solutions, support your answers graphically.

$$x^2=-32$$

One or more zeros are given for this polynomial. Find all remaining zeros.

$P(x)=2 x^4-2 x^3+55 x^2-50 x+125 ; \quad-5 i$ is a zero.

Determine whether the given statement is true or false. If it is false, tell why.

A complex number might not be a pure imaginary number.

Use an end behavior diagram (the ones shown in the discussion) to describe the end behavior of the graph of each function. Do not use a calculator.

$$P(x)=-x^{104,266}$$

Solve the given problem.

A cylindrical aluminum can is being constructed to have a height $h$ of four inches. If the can is to have a volume of twenty-eight cubic inches, approximate its radius $r$. (Hint: $V=\pi r^2 h$.)

In this exercise, graph the function in the standard window of a calculator, and use the root-finding capabilities to solve the equation $P(x)=0$. Express solutions as approximations to the nearest hundredth. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$

For the given complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex.

$$3 i$$

Use the intermediate value theorem to show that the provided function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth.

$P(x)=2 x^7-x^4+x-4 ; \quad 1.1$ and $1.2$

Match the given equation in Column I with its solution(s) in Column II. Do not use a calculator.

$$\begin{array}{lll} &\text{I}&\text{II}\\ & x^2=-4 &\text{A. }\pm 2 i\\ & &\text{B. } \pm 2 \sqrt{2}\\ & &\text{C. }\pm i \sqrt{2}\\ & &\text{D. }2\\ & &\text{E. }\pm \sqrt{2}\\ & &\text{F. }-2\\ & &\text{G. }\pm 2\\ & &\text{H. }\pm 2 i \sqrt{2}\\ \end{array}$$

In this exercise, use the discriminant to explain how to determine the number of $x$-intercepts the graph of $P$ will have before graphing it on your calculator. Consider the function

$$P(x)=-2.64 x^2+5.47 x+3.54 .$$