Exam 3: Polynomial and Rational Functions

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. $4 + 1 i , - 9 + i \text {; degree } 4$

Salt water of concentration 0.5 pound of salt per gallon flows into a large tank that initially contains 220 gallons of pure water. If the flow rate of salt water into the tank is 4 gal/min, find a formula for the salt concentration $f ( \mathrm { t } )$ (in lb/gal) after t minutes.

Use synthetic division to decide whether $c$ is a zero of the equation. $f ( x ) = 7 x ^ { 4 } + 30 x ^ { 3 } - 4 x ^ { 2 } - 46 x + 9$ ; $c = - 4$

The pressure P acting at a point in a liquid is directly proportional to the distance d from the surface of the liquid to the point. Express P as a function of d by means of a formula that involves a constant of proportionality k. In a certain oil tank, the pressure at a depth of 8 feet is 472. Find the value of k.

Use synthetic division to find $f ( c )$ $f ( x ) = 5 x ^ { 3 } + 6 x ^ { 2 } - 8 x + 5$ ; $c = 3$

Find all solutions of the equation. $x ^ { 3 } - x ^ { 2 } - 24 x + 36 = 0$

The density D(h) (in kg/m ³ ) of the earth's atmosphere at an altitude of h meters can be approximated by $D ( h ) = 1.2 - a h + b h ^ { 2 } - c h ^ { 3 } \text {, }$ where $a = 1.096 \times 10 ^ { - 4 } , b = 3.42 \times 10 ^ { - 9 } , c = 3.6 \times 10 ^ { - 14 } ,$ and $0 \leq h \leq 30,000$ . Use a graphing utility to graph D and approximate the altitude h at which the density is 0.3.

Find the quotient and remainder if $f ( k )$ is divided by $p ( k )$ $f ( x ) = 8 x + 3$ $p ( x ) = 5 x ^ { 2 } - x - 7$

Find the fourth-degree polynomial function whose graph is shown in the figure.

Poiseuille's law states that the blood flow rate F ( in L/min ) through a major artery is directly proportional to the product of the fourth power of the radius r and the blood pressure P. During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by 7%, approximately how much harder must the heart pump?

Find a factored form with integer coefficients of the polynomial f shown in the figure. $f ( x ) = 10 x ^ { 5 } - 41 x ^ { 4 } + 52 x ^ { 3 } - 17 x ^ { 2 } - 8 x + 4$

Find all values of $X$ such that $f ( x ) > 0$ $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 16 x - 64$

Find all values of $X$ such that $f ( x ) > 0$ $f ( x ) = x ^ { 4 } - 16 x ^ { 2 }$

A canvas camping tent is to be constructed in the shape of a pyramid with a square base. An 8-foot pole will form the center support, as illustrated in the figure. Find the length x of a side of the base so that the total amount of canvas needed for the sides and bottom is 384 ft ² .

Express the statement as a formula that involves the variables w, z, u and a constant of proportionality k, and then determine the value of k from the condition : w varies directly as z and inversely as the square root of u, if z = 3 and u = 4, then w = 18

Express the statement as a formula that involves the variables q, x, y and a constant of proportionality k, and then determine the value of k from the condition : q is inversely proportional to the sum of x and y, if x = 2.5 and y = 3.6, then q = 3.8

When uranium disintegrates into lead, one step in the process is the radioactive decay of radium into radon gas. Radon enters through the soil into home basements, where it presents a health hazard if inhaled. In the simplest case of radon detection, a sample of air with volume V is taken. After equilibrium has been established, the radioactive decay D of the radon gas is counted with efficiency E over time t. The radon concentration C present in the sample of air varies directly as the product of D and E and inversely as the product of V and t. For a fixed radon concentration C and time t, find the change in the radioactive decay count D if V is multiplied by 2 and E is reduced by 14%.

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zero and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R. $7 + 2 i \text {; degree } 2$

Find all values of $X$ such that $f ( x ) > 0$ $f ( x ) = x ^ { 2 } ( x + 4 ) ( x - 3 ) ^ { 2 } ( x - 4 )$ .

From a rectangular piece of cardboard having dimensions W = 18 and L = 40 an open box is to be made by cutting out identical squares of area x ² from each corner and turning up the sides (see Illustration). Find all positive values of x such that the volume of the box V ( x ) > 0. Include only allowable values of x in your answer.