Exam 4: Polynomial and Rational Functions

Solve the problem -A rectangular piece of cardboard measuring 15 inches by 45 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 $\mathrm { x }$ represent the length of a side of each such square. What is the maximum volume of this box? If necessary, round to 2 decimal places.

For the polynomial, one zero is given. Find all others. - $P ( x ) = x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 ; 1 + i$

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given. - $f ( x ) = 3 x ^ { 5 } + 4 x ^ { 3 } - 9 x ^ { 2 } + 2 ; - 1$ and 0

Round your answer to two decimal places. -According to Ohm's law, the electric current I, in amperes, in a circuit varies directly as the voltage V. When 25 volts are applied, the current is 4 amperes. What is the current when 7 volts are Applied?

Give all possible rational zeros for the following polynomial. - $P ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } + 12 x - 8$

Use the equation and the corresponding graph for the quadratic function to find what is requested. - $f(x)=-2(x-2)^{2}+1$ Find the domain and range.

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

Round your answer to two decimal places. -The intensity I of light varies inversely as the square of the distance $D$ from the source. If the intensity of illumination on a screen $5 \mathrm { ft }$ from a light is 3 foot candles, find the intensity on a screen $15 \mathrm { ft }$ from the light.

Provide an appropriate response. -For what value of $c$ does the quadratic function $f ( x ) = x ^ { 2 } - 6 x + c$ have exactly one $x$ -intercept?

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. - $k = - 3 ; f ( x ) = x ^ { 4 } + 5 x ^ { 2 } - 126$

Sketch the graph of the rational function. - $f ( x ) = \frac { x ^ { 2 } - 4 } { x - 2 }$

2Given the equation or other information for a parabola, find the matching description or graph. - f(x)=a+bx+c a>0;-4ac>0

Find the correct end behavior diagram for the given polynomial function. - $P ( x ) = 6 x ^ { 6 } - x ^ { 5 } + 8 x ^ { 2 } - 7$

Find an equation for the rational function graph. -

Give the equation of any oblique asymptotes. - $h ( x ) = \frac { 7 x - 6 } { 8 x ^ { 2 } - 3 x + 2 }$

Provide an appropriate response. -The domain of a polynomial function is the set of all real numbers. What is the range of a polynomial function of odd degree? One of even degree?

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k. - $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 10 x ^ { 2 } + 15 ; \quad k = 2$

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given. - $f ( x ) = 7 x ^ { 3 } + 9 x ^ { 2 } - 7 x + 4 ; - 2$ and $- 1$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 3 x ^ { 6 } - 7 x ^ { 4 } - 7 x ^ { 3 } + 4 x ^ { 2 } - 5 x$

Provide an appropriate response. -Explain the behavior of the graph of $f ( x )$ as it approaches its vertical asymptote. $f ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }$