Give a short written answer in this exercises.

The graphs of $f(x)=x^n$ for $n=3,5,7, \ldots$ resemble each other. As $n$ gets larger, what happens to the graph?

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

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

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.

$$P(x)=12 x^{107,4 y y}$$

Solve this equation analytically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as $\mathrm{Y}_1$ in the suggested viewing window and, using the capabilities of your calculator, support the real solutions.

$4 x^4-29 x^2+25=0$; $[-5,5]$ by $[-50,100]$

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.

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 $6+2 i$

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

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

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.

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