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

$$(4 x+1)^2=20$$

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

$1+\sqrt{2}, 1-\sqrt{2}$, and $3$

Solve the given problem.

A piece of sheet metal is two and a half times as long as it is wide. It is to be made into a box with an open top by cutting three-inch squares from each corner and folding up the sides, as shown below. Let $x$ represent the width of the original piece of sheet metal.

(a) Represent the length of the original piece of sheet metal in terms of $x$.

(b) What are the restrictions on $x$?

(c) Determine a function $V$ that represents the volume of the box in terms of $x$.

(d) For what values of $x$ (that is, original widths) will the volume of the box be between $600$ and $800$ cubic inches? Determine the answer graphically, and give values to the nearest tenth of an inch.

Write this complex number in standard form. Let $w=17 -i$ und let $z=1-3i$.

$$w+z$$

Write the given number in simplest form, without a negative radicand.

$$\sqrt{-169}$$

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

$-3,2$, and $i$

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

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

Solve the given problem.

A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting two-inch squares from each corner and folding up the sides. Let $x$ represent the width of the original piece of cardboard.

(a) Represent the length of the original piece of cardboard in terms of $x$.

(b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on $x$.

(c) Determine a function $V$ that represents the volume of the box in terms of $x$.

(d) For what dimensions of the bottom of the box will the volume be $320$ cubic inches? Determine analytically and support graphically.

(e) Determine graphically (to the nearest tenth of an inch) the values of $x$ if the box is to have a volume between $400$ and $500$ cubic inches.

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?

Find the quotient when $P(x)$ is divided by the specified binomial.

$$P(x)=x^3+2 x^2-17 x-10 ;\quad x+5$$

Write the given number in simplest form, without a negative radicand.

$$\sqrt{-100}$$

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

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

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.

$x^4-15 x^2-16=0$; $[-5,5]$ by $[-100,100]$

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