Exam 4: Polynomial and Rational Functions

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. -Zero of $- 4$ having multiplicity 2 and zero of 8 having multiplicity $1 ; P ( 0 ) = - 128$

Identify the vertex of the parabola. - $y = 5 x ^ { 2 } - 50 x + 128$

Round to the nearest tenth unless indicated otherwise. -The resistance of a wire varies directly as the length of the wire and inversely as the square of the diameter of the wire. A 20 foot length of wire with a diameter of $0.1$ inch has a resistance of 3 ohms What would the resistance be for a 21 foot length, with diameter $0.01$ inch, of the same kind of wire ?

Solve the problem -If an object is thrown upward with an initial velocity of $48 \mathrm { ft } / \mathrm { sec }$ , its height after $\mathrm { t }$ sec is given by $\mathrm { s } ( \mathrm { t } ) = 48 \mathrm { t } - 16 \mathrm { t } ^ { 2 }$ . Find the maximum height attained by the object. (The object will attain maximum height exactly at the halfway point in terms of the time $t$ , where $t = 0$ is at the beginning of the object's flight, and the final time is when the object hits the ground.)

Translate the given formula to an English phrase using the word "varies". - $\mathrm { P } = \frac { \mathrm { kNT } } { \mathrm { V } }$ , where $\mathrm { P }$ is the gas pressure of $\mathrm { N }$ molecules of a gas in a volume $\mathrm { V }$ at temperature $\mathrm { T }$

Use the factor theorem to decide whether or not the second polynomial is a factor of the first. - $5 x ^ { 4 } + 11 x ^ { 3 } - 2 x ^ { 2 } + x + 4 ; x + 2$

Identify the vertex of the parabola. - $f ( x ) = ( x + 4 ) ^ { 2 } - 1$

Solve the problem -A toy rocket is shot vertically upward from the ground. Its distance in feet from the ground in $t$ seconds is given by $\mathrm { s } ( \mathrm { t } ) = - 16 \left( \mathrm { t } ^ { 2 } \right) + 142 \mathrm { t }$ . At what time(s) will the ball be $151 \mathrm { ft }$ from the ground? Round your answer to the nearest tenth of a second.

Graph the function. - $f ( x ) = - \frac { 7 } { x }$

Find the correct end behavior diagram for the given polynomial function. - $P ( x ) = - 2 x ^ { 6 } + 6 x ^ { 5 } - x ^ { 2 } - 6 x + 9$

Solve the problem -A rectangular piece of cardboard measuring 19 inches by 28 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent The length of a side of each such square. For what value of x will the volume be a maximum? If Necessary, round to 2 decimal places.

Find all rational zeros and factor f(x). - $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } - x + 7$

Provide an appropriate response. -Explain the behavior of the graph of f(x) as it approaches its vertical asymptote. $f ( x ) = \frac { - 4 } { ( x - 8 ) ^ { 2 } }$

Factor f(x) into linear factors given that k is a zero of f(x). - $f ( x ) = x ^ { 3 } - 48 x - 128 ; k = - 4$ (multiplicity 2 $)$

Sketch the graph of the parabola. - $y=2 x^{2}-2 x-4$

Tell whether a linear model or a quadratic model is appropriate for the data. If linear, tell whether the slope should be positive or negative. If quadratic, decide whether the leading coefficient a of $x ^ { 2 }$ should be positive or negative. -

Answer the question - $\text { How can the graph of } f ( x ) = \frac { 1 } { x } + 10 \text { be obtained from the graph of } y = \frac { 1 } { x } \text { ? }$

Provide an appropriate response. -Find an equation for a rational function with the following asymptotes: vertical asymptotes x = 5 and x = -2, horizontal asymptote y = 6

Use the remainder theorem and synthetic division to find f(k). - $\mathrm { k } = - 2 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 3 } - 7 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 12$

Solve the problem -The length of a rectangle is 8 inches more than its width. If 4 inches are taken from the length and added to the width, the figure becomes a square with an area of 256 square inches. What are the dimensions of the original figure?