Exam 4: Polynomial and Rational Functions

The height in feet of a projectile with an initial velocity of 128 feet per second and an initial height of 50 feet is a function of time t, in seconds, given by $h ( t ) = - 16 t ^ { 2 } + 128 t + 50.$ Find the maximum height of the projectile.

The rational function $M ( t ) = \frac { 0.5 t + 150 } { 0.04 t ^ { 2 } + 3 }$ models the number of milligrams of medication in the bloodstream of a patient t hours after 150 milligrams of the medication have been injected into the patient's bloodstream. What will M approach as $t \rightarrow \infty$ ?

Simplify the rational expression, $\frac { x ^ { 4 } - 5 x ^ { 3 } - 6 x ^ { 2 } + 32 x + 32 } { x ^ { 2 } - 8 x + 16 }$ , by using long division or synthetic division.

Simplify and write the following complex number in standard form. $( - 6 - 5 i ) ( - 3 + 2 i )$

Find all real solutions of the polynomial equation $x ^ { 4 } - 8 x ^ { 3 } + 56 x - 49 = 0$ .

Find the zeros of the polynomial function below. If a zero is a multiple zero, state its multiplicity. $P ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 15 x + 7$

A company that produces video games estimates that the profit $P$ (in dollars) for selling a new game is given by $P = - 82 x ^ { 3 } + 7250 x ^ { 2 } - 450,000,$ $0 \leq x \leq 80$ where $x$ is the advertising expense (in tens of thousands of dollars). Using this model, how much should the company spend on advertising to obtain a profit of $6,100,000? Round to the nearest dollar.

Given $f ( x ) = \frac { x ^ { 2 } - 8 x + 16 } { x - 1 }$ , determine the equations of any slant and vertical asymptote.

The graph of the function $f ( x ) = \frac { 3 } { x - 2 }$ is shown below. Determine the domain.

An open box is to be made from a square piece of cardboard, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). After determining the function V, in terms of x, that represents the volume of the box, use a graphing utility to estimate the dimensions that will maximize its volume.

Determine the vertical and slant asymptotes of the rational function below. $F ( x ) = \frac { x ^ { 3 } - 9 } { x ^ { 2 } - 4 }$

Simplify $( 5 + 2 i ) ^ { 2 } - ( 5 - 2 i ) ^ { 2 }$ and write the answer in standard form.

The demand and cost equations for a stethoscope are given by $p = 134 - 0.0002 x$ and $C = 50 x + 160,000$ where $p$ is the unit price (in dollars), $C$ is the total cost (in dollars), and $x$ is the number of units. The total profit $P$ (in dollars) obtained by producing and selling $x$ units is given by $P = R - C = x p - C.$ Determine a price $p$ that would yield a profit of $6.9 million.

Use synthetic division to divide. $\left( 5 x ^ { 3 } - 8 x ^ { 2 } - 7 x + 6 \right) \div ( x - 2 )$

Find an equation of the parabola that has a vertex at $( - 5 , - 7 )$ and whose graph passes through the point $( - 6 , - 10 ).$

Find the domain of $f ( x ) = \frac { x - 6 } { x ^ { 2 } - 36 }$ .

Match the equation with its graph. $f ( x ) = \frac { 1 } { 20 } \left( x ^ { 5 } - x ^ { 4 } - 5 x ^ { 3 } - x ^ { 2 } - 6 x \right)$

Write the polynomial $x ^ { 4 } - 6 x ^ { 2 } - 72$ as the product of linear and quadratic factors that are irreducible over the reals.

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 5 x ^ { 2 } + 4 x - 5$ .

The fuel efficiency, in miles per gallon, for a certain midsize car at various speeds, in miles per hour, is given in the table below. 25 20 55 31 30 24 60 33 35 32 65 30 40 35 70 24 45 36 75 22 50 40 Find a quadratic model for these data and use it to predict the fuel efficiency of this car when it is traveling at a speed of 40 mph. Do not round any values in your calculations but round the final answer to the nearest tenth.