Exam 3: Polynomial and Rational Functions

Evaluate the expression $- ( 1 - \sqrt { - 2 } ) + ( 1 - ( \sqrt { - 1 } - \sqrt { - 2 } ) )$ and write the result in the form $a + b i$ .

Find the quotient and remainder of $\frac { x ^ { 4 } - 4 x ^ { 2 } - 2 x ^ { 3 } + 13 x } { x ^ { 2 } - 4 }$ using long division.

A grain silo consists of a cylindrical main section with hemispherical roof. If the total volume of the silo (including the part inside the roof section) is $28,000$ ft $3$ , and the total height of the silo up to the top of the roof is $55$ ft, what is the approximate radius of the silo?

Find the maximum or the minimum value of the function $f ( x ) = - \frac { x ^ { 2 } } { 7 } + 3 x + 5$ .

Find the intercepts and asymptotes, and then graph the rational function $\gamma ( x ) = \frac { x - 1 } { ( x + 1 ) ^ { 2 } }$ .

Evaluate the expression and write the result in the form a + bi. $3 i \left( \frac { 1 } { 3 } - i \right)$

Find all zeros of the polynomial $P ( x ) = x ^ { 4 } - 1$ .

Find the quotient and remainder of $\frac { x ^ { 3 } + 8 x + 4 } { 3 x - 2 }$ using synthetic division.

Find all rational zeros of the polynomial $p ( x ) = 4 x ^ { 5 } + 24 x ^ { 4 } + 53 x ^ { 3 } + 53 x ^ { 2 } + 24 x + 4$ .

An open box is to be constructed from a piece of sheet metal $15$ in by $25$ in by cutting squares of length $x$ from each corner and folding up the sides. (a) Express the volume $V$ of the box as a function of $x$ . (b) What is the domain of $V$ ? (c) Draw a graph of the function $V$ and use it to estimate the maximum volume for the box.

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function, $R ( x ) = 40 x - 0.2 x ^ { 2 }$ 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?

Find a polynomial of degree 3 that has zeros 1, -2, and 3 and in which the coefficient of $x ^ { 2 }$ is -5.

A business consultant analyzing a small computer company finds that if the company produces and sells $x$ microprocessors it has a total profit function (in thousands of dollars) given by $p ( x ) = 0.4 x ^ { 2 } - 0.0043 x ^ { 3 } + 7 x - 244$ , with $x$ (in hundreds), being the number of processors produced and sold annually. (a) Graph the profit function and determine how many processors must be produced and sold in order to break even. (b) What is the domain of the profit function where the company actually loses money? (c) Estimate the number of processors that must be produced in order to give the company the largest possible profit. (d) Estimate the largest profit.

Sketch the graph of the function $y = x ^ { 4 } + 2 x ^ { 2 } - 1$ and describe its end behavior.

Find the quotient and remainder of $\frac { x ^ { 4 } - 2 x ^ { 3 } - x + 2 } { x + 2 }$ using synthetic division.

Find all real zeros of the polynomial $p ( x ) = x ^ { 4 } - 9 x ^ { 2 } + 20$ . Use the quadratic formula if necessary.

Find all real zeros of the polynomial $p ( x ) = - x ^ { 6 } + 10 x ^ { 4 } - 25 x ^ { 2 }$ . Sketch the graph of $p$ .

Find all rational zeros of the polynomial $p ( x ) = 18 x ^ { 4 } - 33 x ^ { 3 } - 7 x ^ { 2 } + 32 x - 12$ .

Let $f ( x ) = c x ^ { 4 }$ . Graph the family of functions with $c = \frac { 1 } { 4 }$ , $\frac { 1 } { 2 }$ , $1$ , $2$ , and $4$ in the viewing rectangle $[ - 2,2 ]$ by $[ - 1,8 ]$ . How does the value of $c$ affect the graph?

Find the quotient and remainder of $\frac { x ^ { 4 } - 256 } { x + 4 }$ using synthetic division.