Exam 3: Polynomial and Rational Functions

Find all asymptotes (including vertical, horizontal, and slant) of the function $y = \frac { 2 x + 4 } { x - 2 }$ .

Find all rational zeros of the polynomial $p ( x ) = x ^ { 6 } + 2 x ^ { 5 } - 7 x ^ { 4 } - 20 x ^ { 3 } - 12 x ^ { 2 }$ .

Sketch the graph of the function $p ( x ) = - \left( x ^ { 6 } - 5 x ^ { 3 } + 6 \right)$ . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Find all real zeros of the polynomial $p ( x ) = x ^ { 5 } - 4 x ^ { 4 } - 13 x ^ { 3 } + 70 x ^ { 2 } - 90 x + 36$ . Sketch the graph of $p$ .

Sketch the graph of the function $y = - 2 x ^ { 5 } + 2$ by transforming the graph of an appropriate function of the form $y = x ^ { n }$ . Indicate all $x$ - and $y$ - intercepts on your graph.

Find all real zeros of the polynomial $p ( x ) = x ^ { 3 } - 6 x - 4$ . Use the quadratic formula if necessary.

Evaluate the expression and write the result in the form a + bi. $( 2 - 8 i ) + ( 3 + 7 i )$

A drug is administered to a patient and the concentration $c$ of the drug in the bloodstream is monitored. At the time $t \geq 0$ (in minutes since the injection), the concentration (in mg/L) is given by $c ( t ) = \frac { 55 t } { t ^ { 2 } + 3 }$ . (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of the drug in the bloodstream? (c) What is the highest concentration of the drug that is reached in the patient's bloodstream? When does this occur?

Sketch the graph of the function $y = x ^ { 6 } + 2$ by transforming the graph of an appropriate function of the form $y = x ^ { n }$ . Indicate all $x$ - and $y$ - intercepts on your graph.

On the same coordinate axes, sketch the graphs (as accurately as possible) of the functions $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3$ and $g ( x ) = - 4 x ^ { 2 } + 3 x + 3$ . Based on your sketch, how many points do the graphs appear to intersect? Find the coordinates of all intersection points.

Find all solutions of the equation $\frac { t - 1 } { 3 } + 3 + \frac { 5 } { t + 1 } = 0$ and express them in standard form $a + b i$

Find a polynomial $P$ of degree $2$ , with integer coefficients that has a zero of $1 - 2 i$ .

Find intercepts and asymptotes, and then graph the rational function $y = \frac { x ^ { 3 } - x ^ { 2 } - 4 x + 4 } { 2 x ^ { 3 } + 3 x }$ .

Find all horizontal and vertical asymptotes of the function $r ( x ) = \frac { 5 } { x - 5 }$ .

Graph the family of polynomials $P ( x ) = ( x + c ) ^ { 4 }$ in the same viewing rectangle, using the values $c = - 2$ , $- 1$ , $0$ , $1$ , and $2$ . Explain how changing the value of $c$ affects the graph.

Find a polynomial $P$ of degree $5$ , with integer coefficients, that has zeros of $i$ and $2$ , with $i$ being a zero of multiplicity $2$ , and $P$ having a leading coefficient of $2$ .

Use the Factor Theorem to show that $( x - 4 )$ is a factor of the polynomial $P ( x ) = x ^ { 5 } + x ^ { 4 } - 36 x ^ { 3 } - 16 x ^ { 2 } + 320 x$ .

Find all zeros of the polynomial $P ( x ) = x ^ { 5 } + 8 x ^ { 3 } + 16 x - x ^ { 4 } - 8 x ^ { 2 } - 16$ .

Find the quotient and remainder of $\frac { x ^ { 3 } + 8 x + 24 } { x ^ { 2 } - 3 x + 6 }$ using long division.

Find the quotient and remainder using long division. $\frac { x ^ { 6 } + x ^ { 2 } + 9 x ^ { 4 } + 9 } { x ^ { 2 } + 9 }$