Exam 4: Polynomials and Rational Functions

A farmer has 400 feet of fence with which to fence a rectangular plot of land. The plot lies along a river so that only three sides need to be fenced. Estimate the largest area that can be fenced.

Find the domain and range of the function. $f ( x ) = x ^ { 2 } + 2 x - 4$

Use synthetic division to perform the division. $\frac { 2 x ^ { 3 } + x ^ { 2 } - 3 x + 2 } { x + 3 }$

Give the equations of any horizontal asymptotes. - $g ( x ) = \frac { x ^ { 2 } + 8 x - 1 } { x - 1 }$

A rug is to fit in a room so that a border of even width is left on all four sides. If the room is 12 feet by 15 feet and the area of the rug is 108 square feet, how wide will the border be?

The table shows the population of a city over the past five years. We used these data to develop the quadratic function $f ( x ) = 0.037214 x ^ { 2 } + 0.364 x + 65$ , which models the populatic of the city $y$ in millions in the year $x$ . Use the model to find the estimated population in year $10 .$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 + i , 4$

Identify the vertex of the parabola. - $f ( x ) = - 6 ( x + 8 ) ^ { 2 } - 7$

Find the equation that the given graph represents. -

Determine whether the statement is true or false. - $f(x)=\frac{1}{x}-4$

The data in the table show the number of burglaries reported in a small town for several years on record. Determine the linear, quadratic, cubic, or quartic function that best fits the data.

Round to the nearest tenth unless indicated otherwise. -The volume $\mathrm { V }$ of a given mass of gas varies directly as the temperature $\mathrm { T }$ and inversely as the pressure $\mathrm { P }$ . If $\mathrm { V } =$ $180.0$ in $^ { 3 }$ when $\mathrm { T } = 300 ^ { \circ }$ and $\mathrm { P } = 20 \mathrm { lb } / \mathrm { in } ^ { 2 }$ , what is the volume when $\mathrm { T } = 210 ^ { \circ }$ and $\mathrm { P } = 15 \mathrm { lb } / \mathrm { in } ^ { 2 }$ ?

Sketch the graph of the parabola. -

Give all possible rational zeros for the following polynomial. $P ( x ) = 14 x ^ { 3 } + 56 x ^ { 2 } + 2 x - 7$

Your company uses the quadratic model y $y = - 11 x ^ { 2 } + 350 x$ represent how many units (y) of a new product will be sold (x) weeks after its release. How many units can you expect to sell in week 24?

For the polynomial, one zero is given. Find all others. - $P ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 11 x + 52 ; - 4$

Use synthetic division $f ( x ) \text { by } x - k \text { for the given value of } k \text {. Then express } f ( x ) \text { in the form } f ( x ) = ( x - k ) q ( x ) + r$ for the given value of k. - $f ( x ) = x ^ { 3 } - x ^ { 2 } + 6 ; k = - 2$

Find a polynomial of least degree with only real coefficients and having the given zeros. - $2 - i , - 6$

Round to the nearest tenth unless indicated otherwise. -The volume of a gas varies inversely as the pressure and directly as the temperature (in degrees Kelvin). If a certain gas occupies a volume of 2.2 liters at a temperature of 340 K and a pressure of 16 newtons per square Centimeter, find the volume when the temperature is 306 K and the pressure is 24 newtons per square Centimeter.

The following polynomial approximates the shark population in a particular area. $R ( x ) = - 0.021 x ^ { 5 } + 4.133 x ^ { 4 } + 200$ where x is the number of years from year 0. Use a graphing calculator to describe the shark population from the years 0 to 25.