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

$-3,2$, and $i$

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

$$x^2=-32$$

Answer the following given problems.

Suppose that a polynomial function $P$ is defined in such a way that $P(3)=-4$ and $P(4)=-10$. Can we be certain that there is no zero between $3$ and $4$? Explain, using a graph.

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

$6$ and $-2$

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.

In this exercise, verify analytically that the answer in the previous exercise is correct. Consider the function

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

Find a cubic polynomial in standard form with real coefficients, having the given zeros. Let the leading coefficient be $1$. Do not use a calculator.

$-3$ and $6+2 i$

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: $V=\pi r^2 h$.)

Use $P(x)=2 x^4-7 x^3+29 x-30$. Is $2$ a zero of $P$ ?

For the function $P(x)=-2 x^2-4 x+6$, do the following. (a) Find the vertex analytically. (b) Give a comprehensive graph, and use a calculator to support your result in part (a). (c) Find the zeros of $P$ and support your result, using a graph for one zero and a table for the other. (d) Find the $y$-intercept analytically. (e) State the domain and range of $P$. (f) Give the largest open interval over which the function is increasing and the largest open interval over which it is decreasing.

Find all real solutions. Do not use a calculator.

$$x^4-x^3-6 x^2=0$$

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.

$$3 i$$

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, simplify the given rational expression. Apply the property $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$ if necessary. Do not use a calculator.

$$\frac{6 x^4}{2 x^3}$$