Use the technique of completing the square to write \(f(x)=x^2-8x+4\) in the form \(f(x)=a(x-h)^2+k\).

Find the \(x\)-intercepts of the parabola associated with the function \(f(x)=3x^2-5x-2\). Also, find the real zeros of \(f\).

Find a possible expression for a quadratic function \(f(x)\) with zeros at \(x=-3\) and \(x=1\).

Traffic authorities have \(100\) feet of rope to cordon off a rectangular region to form a ticket arena specifically for concert goers who are waiting to purchase tickets.

1. Express the area of this region as a function of the length of just one of the four sides of the region.

2. Find the dimensions of the enclosed region that will give the maximum area, and determine the maximum area.

The following table lists the speed at which a car is driven, in miles per hour, and the corresponding gas mileage obtained, in miles per gallon.

1. Make a scatter plot of \(m\), the gas mileage, vs. \(x\), the speed at which a car is driven. What type of trend do you observe - linear or quadratic? Find an expression for the function that best fits the given data points.

2. Use your function to find the gas mileage obtained when a car is driven at \(35\) miles per hour.

3. Find the speed at which the car’s gas mileage is at its maximum, using a graphing utility.

Graph the following functions using transformations.

1. \(g(x)=(x-1)^4\)

2. \(g(x)=-x^3+1\)

Find the real zeros of \(f(x)=2x^3-18x\) and the corresponding \(x\)-intercepts of the graph of \(f\).

Check whether \(f(x)=x^4+x^2+x\) is odd, even, or neither.

Find a polynomial \(f(x)\) of degree \(4\) that has zeros at \(-1\) and \(3\), each with multiplicity \(1\), and a zero at \(-2\) of multiplicity \(2\).

Gift Horse, Inc., manufactures various type of decorative gift boxes. The bottom portion of one such box is made by cutting a small square of length \(x\) inches from each corner of a \(10\)-inch piece of cardboard and folding up the sides of the box. See Figure 16.

1. Find an expression for the volume of the resulting box.

2. Graph the volume function. Use your graph to determine the values of \(x\) for which the expression makes realistic sense.

let \(p(x)=-2x^4+6x^3+3x-1\). Use synthetic division to evaluate \(p(2)\).

Fill in Table 1, where \(p\), \(h\), and \(g\) are some polynomial functions.

Let \(h(x)=x^3+2x^2+x\)

1. What is the value of the multiplicity, \(k\), of the zero at \(x=-1\)?

2. Write \(h(x)\) in the form \(h(x)=(x+1)^kQ(x)\). What is \(Q(x)\)?

Find all zeros \(p(x)=x^3-x^2-3x\) and factor \(p(x)\) over the complex numbers.

Factor \(f(x)=2x^5+12x^3+18x\) ever the complex numbers.

Find a polynomial \(p(x)\) of degree \(4\) with \(p(0)=-9\) and zeros \(x=-3\), \(x=1\), and \(x=3\), with \(x=3\) a zero of multiplicity \(2\). For this polynomial, is it possible for the zeros other than \(3\) to have a multiplicity greater than \(1\)?

Factor \(f(x)=x^4-2x^3-3x^2-2x+6\) over the complex numbers, and find all zeros.

Write all vertical asymptotes of the following functions.

1. \(f(x)=\dfrac{2x}{x+1}\)

2. \(f(x)=\dfrac{x+2}{x^2-1}\)

3. \(f(x)=\dfrac{x^2-3}{2x+1}\)

Find the horizontal asymptote, if it exists, for each of the following rational functions. Use a table and a graph to discuss the end behavior of each function.

1. \(f(x)=\dfrac{x+2}{x^2-1}\)

2. \(f(x)=\dfrac{2x}{x+1}\)

3. \(f(x)=\dfrac{x^2-3}{2x+1}\)

Sketch the graph of \(f(x)=\dfrac{x+2}{x^2-1}\)

Sketch the complete graph of \(r(x)=\dfrac{x-1}{x^2-3x+2}\)

Suppose it costs \(\$45\) a day to rent a car with unlimited mileage.

1. What is the expression for the average cost per mile per day?

2. Make a table and graph of the average cost function.

3. What happens to the average cost per day as the number of miles driven per day increases?

Solve the inequality \(\dfrac{2x+5}{x-2}\geq x\)