Exam 4: Polynomial and Rational Functions

The cost, in dollars, of producing x golf balls is given by $C ( x ) = 0.44 x + 74,000.$ The average cost per golf ball is given by $\bar { C } ( x ) = \frac { C ( x ) } { x } = \frac { 0.44 x + 74,000 } { x }.$ Find the average cost of producing 100,000 golf balls.

Sketch the graph of the rational function. Check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $h ( x ) = - \frac { x } { x ^ { 2 } - 9 }$

Using the factors $( 3 x + 2 )$ and $( x - 1 )$ , find the remaining factor(s) of $f ( x ) = - 6 x ^ { 4 } + 23 x ^ { 3 } - 12 x ^ { 2 } - 11 x + 6$ and write the polynomial in fully factored form.

Simplify and write the following complex number in standard form. $( - 6 - i ) + ( 5 - 10 i )$

Use the zero or root feature of a graphing utility to approximate the real zeros of the function $f ( x ) = x ^ { 6 } - 4 x ^ { 4 } + x ^ { 2 } + 6$ accurate to the nearest thousandth.

Write $f ( x ) = x ^ { 5 } - 6 x ^ { 4 } + 15 x ^ { 3 } - 18 x ^ { 2 } + 10 x$ as a product of linear factors given that $2 - i$ and $1 - i$ are zeros of the polynomial.

The path of a diver is given by $y = - \frac { 16 } { 25 } x ^ { 2 } + \frac { 24 } { 5 } x + 1$ where $y$ is the height (in feet) and $x$ is the horizontal distance from the end of the diving board (in feet). Find the maximum height of the diver.

Use the zero or root of graphing utility to find the real zeros of the function $f ( x ) = x ^ { 4 } = 3 x ^ { 2 } - 5 x ^ { 2 } - 21 x + 22$ and find the exact values of the remaining zeros.

The demand for a stethoscope is given by where p is the unit price ( in dollar) and x is the number of units. The total revenue R (in dollars) obtained by producing ans selling x units given by the model $R = x p = - 0.001 x ^ { 2 } + 140 x$ . Write and solve an equation to determine a price that yields revenue of $50 million. Use a graphing utility to verify your result.

The sum of the length l and the width w of a rectangular region is 210 meters. Write the area A as a function of w.

Use long division to divide. $\left( x ^ { 3 } - 5 x ^ { 2 } + 9 x - 45 \right) \div ( x - 5 )$

Write the complex number $12 + \sqrt { - 16 }$ in standard form.

Use the zero $5 + 2 i$ of $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } - x + 87$ to find all the zeros of f.

Find all real solutions of the polynomial equation $6 x ^ { 6 } - x ^ { 5 } + 29 x ^ { 4 } - 5 x ^ { 3 } + 19 x ^ { 2 } - 4 x - 4 = 0$ .

Simplify the rational expression, $\frac { - 4 x ^ { 3 } + 10 x ^ { 2 } + 28 x - 16 } { - 4 x + 2 }$ , by using long division or synthetic division.

Use the zero $\frac { 1 } { 5 } ( - 2 + \sqrt { 2 } i )$ of $f ( x ) = 25 x ^ { 3 } - 55 x ^ { 2 } - 54 x - 18$ to find all the zeros of f.

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } + 9 }$ .

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 41 x ^ { 2 } + 400$ and determine the mutiplicity of each.

Use the Rational Zero Test to list possible rational zeros for the polynomial function below. $P ( x ) = x ^ { 3 } - 7 x ^ { 2 } - 6 x + 20$

Classify whether the following numbers are in the Mandelbrot Set. I : $- 1$ II : $- i$ III : $3 i$