Exam 3: Polynomial and Rational Functions

Assume that the rat population in a certain urban area behaves according to the function $P ( t ) = \frac { 25 t } { t + 1 }$ where $P$ is the number of rats in the hundreds, and $\text { t }$ is measured in months since the beginning of the year.(a) Draw a graph of the rat population. (b) What eventually happens to the rat population?

Find all solutions of the equation $y + 4 + \frac { 5 } { y } = 0$ and express them in standard form $a + b i$

Find a quadratic function whose graph is a parabola with $x$ - intercepts of $( 1,0 )$ and $\left( \frac { 4 } { 3 } , 0 \right)$ and lowest point $( h , - 4 )$ .

Find all solutions of the equation $x ^ { 2 } + 5 x + 25 = 0$ and express them in standard form $a + b i$ .

Find intercepts and asymptotes, and then graph the rational function $y = \frac { 5 x ^ { 3 } - 15 } { 2 x ^ { 3 } + x ^ { 2 } - 50 x - 25 }$ .

Find all zeros of the polynomial $P ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } + 14 x + 12 = ( 2 x + 3 ) \left( x ^ { 2 } + 2 x + 4 \right)$ .

Find the complete factorization of $P ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 2 x + 1$

If a ball is thrown directly upward with a velocity of 48 ft/s, its height (in feet) after t seconds is given by $y = 48 t - 16 t ^ { 2 }$ . What is the maximum height attained by the ball?

Find all horizontal and vertical asymptotes of the function $s ( x ) = \frac { x ^ { 3 } + 6 x ^ { 2 } } { x ^ { 2 } - 25 }$ .

Find all horizontal and vertical asymptotes of the function $\gamma ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 11 x + 30 }$ .

Find the quotient and remainder of $\frac { x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } - x + 5 } { x ^ { 2 } - 1 }$ .

Find all rational zeros of the polynomial $p ( x ) = 9 x ^ { 4 } + 30 x ^ { 3 } + 13 x ^ { 2 } - 20 x + 4$ .

Evaluate the expression $\frac { 1 } { \sqrt { - 1 } }$ and write the result in the form $a + b i$ .

Find intercepts and asymptotes, and then graph the rational function $y = \frac { x ^ { 3 } - 5 } { x ^ { 2 } - 5 }$ .

The cost associated with storing $x$ units of a certain product is given by the function $C ( x ) = 0.8 x ^ { 2 } - 12.8 x + 2600$ where $C$ is measured in of dollars and $x$ in hundreds of units. (a) For what values of $x$ will the cost be decreasing? increasing? (b) What is the minimum cost?

Use a graphing device to graph the polynomial $y = 3 x ^ { 4 } - 1$ and describe its end behavior.

Find all real zeros of the polynomial $p ( x ) = - x ^ { 3 } + 9 x ^ { 2 } - 24 x + 16$ . Sketch the graph of $p$ .

Find all real zeros of the polynomial $p ( x ) = x ^ { 5 } + 3 x ^ { 4 } + 2 x ^ { 3 } - 2 x ^ { 2 } - 3 x - 1$ . Sketch the graph of $p$ .

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, $R ( x ) = 80 x - 0.2 x ^ { 2 }$ where the revenue $R ( x )$ is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Evaluate the expression $1 + \frac { \sqrt { - 36 } \cdot \sqrt { - 49 } i } { \sqrt { 21 } }$ and write the result in the form $a + b i$ .