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.

Perform the operation. Write answers in standard form.

$$i(5+i)(5-i)$$

Do not use a calculator.

Find the maximum $y$-value on the graph of $y=-2 x^2+8 x-5$.

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

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

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

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

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

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.

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

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.

$$3 i$$

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

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

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 $6+2 i$

Find all real solutions. Do not use a calculator.

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

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.

Research Funding The table gives National Science Foundation funding at the beginning of each decade since 1951. In the table, years are given as the number of years since 1900 . Thus, $x=51$ corresponds to $1951, x=61$ corresponds to 1961 , and so on.

$$\begin{array}{l|c|c|c|c|c|c} \text { Year } x & 51 & 61 & 71 & 81 & 91 & 101 \\ \hline \begin{array}{l} \text { Dollars } y \\ \text { (in billions) } \end{array} & 0.1 & 0.2 & 0.6 & 1.1 & 2.3 & 4.7 \end{array}$$

(a) Plot the data. (b) Find a function $f(x)=a(x-h)^2+k$ that models the data by using $(51,0.1)$ as the vertex and $(101,4.7)$ as the second point. (c) Use the quadratic regression feature of a graphing calculator to determine the quadratic function $g$ that best fits the data. (d) Plot $f$ and $g$ in the same window as the data. Discuss how well the models fit the data.