Exam 4: Polynomial and Rational Functions

A music company expects that the monthly sales S, in thousands, of a new music CD it has produced will be closely approximated by $S ( t ) = \frac { 140 t } { 2.25 t ^ { 2 } + 80 }$ where t is the number of months after the CD is released. Find the monthly sales the company expects 2 months after the CD is released. Round to the nearest 100 CDs.

Combine $( 9 + \sqrt { - 3 } ) ( 8 - \sqrt { - 21 } )$ and write the answer in standard form.

Use long division to divide the first polynomial below by the second. $x ^ { 5 } + x ^ { 4 } + 7 x ^ { 3 } + x ^ { 2 } + 6 x - 27,$ $x ^ { 2 } + 6$

Find all the rational zeros of the function $f ( x ) = - 3 x ^ { 4 } + 4 x ^ { 3 } + 52 x ^ { 2 } - 64 x - 64$ .

Use synthetic division to divide. $\left( 8 + 5 x ^ { 3 } + 46 x + 31 x ^ { 2 } \right) \div ( x + 2 )$

Write the following expression as a complex number in standard form. $\frac { - 4 - 7 i } { - 5 - 8 i }$

The length of a rectangular box is 1 inch more than twice the height of the box, and the width is 3 inches more than the height. If the volume of the box is 440 cubic inches, find the dimensions of the box.

Evaluate the power of i. $i ^ { 236 }$

Write the polynomial $x ^ { 4 } - 7 x ^ { 2 } - 8$ as the product of factors that are irreducible over the rationals.

Compare the graph of $p ( x ) = 4 ( x - 3 ) ^ { 2 } + 9$ with $p ( x ) = x ^ { 2 }$ .

The fuel efficiency, in miles per gallon, for a certain midsize car at various speeds, in miles per hour, is given in the table below. 25 20 55 31 30 24 60 33 35 32 65 30 40 35 70 24 45 36 75 22 50 40 Find a quadratic model for these data.

Plot the complex number below. $- 3 - 2 i$

Use the zero $- 3 + \sqrt { 2 } i$ of $f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 21 x + 22$ to find all the zeros of f.

Simplify $( 2 + i ) ( - 9 + 7 i )$ and write the answer in standard form.

Write $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } - x ^ { 2 } - 38 x + 130$ as a product of linear factors given that $3 - i$ and $- 2 - 3 i$ are zeros of the polynomial.

Use synthetic division to divide $x ^ { 3 } + 4 x ^ { 2 } + 4 x + 3$ by $x - 8$ .

A farmer wants to enlarge an existing rectangular pig pen such that the total area of the new pen is twice as large as the original pen. The current pen's dimensions are 21' × 32'. The farmer wants to add 9 feet more to the long side than to the the short side. Find the dimensions of the new pen.

Determine the vertical and slant asymptotes of the rational function below. $F ( x ) = \frac { x ^ { 2 } + 5 x + 4 } { x + 3 }$

The sum of the length l and the width w of a rectangular region is 240 meters. Find the dimensions that produce the greatest area.

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 11 x ^ { 3 } + 30 x ^ { 2 }$ and determine the mutiplicity of each.