Exam 4: Polynomial and Rational Functions

Solve. -There are $n$ people in a room. The number $N$ of possible handshakes by all the people in the room is given by the function $N ( n ) = \frac { n ( n - 1 ) } { 2 }$ . For what number n of people is $91 \leq N \leq 190$ ?

For the function find the maximum number of real zeros that the function can have, the maximum number of x-intercepts that the function can have, and the maximum number of turning points that the graph of the function can have. - $f ( x ) = - x ^ { 6 } + 8 x ^ { 5 } - x ^ { 2 } - 4 x + 9$

Match the equation with the appropriate graph. - $f ( x ) = \frac { 6 x } { x ^ { 2 } - 1 }$

Use synthetic division to find the function value. - $f ( x ) = x ^ { 5 } - 10 x ^ { 4 } + 15 x ^ { 3 } - 4 x - 300 ;$ find $f ( 2 )$

List the critical values of the related function. Then solve the inequality. - $\frac { 3 } { x ^ { 2 } - 4 x + 3 } \leq \frac { 5 } { x ^ { 2 } - 9 }$

Given that the polynomial function has the given zero, find the other zeros. - $f ( x ) = x ^ { 4 } - 9 ; \sqrt { 3 }$

List the critical values of the related function. Then solve the inequality. - $\frac { x } { x ^ { 2 } + 3 x - 4 } + \frac { 2 } { x ^ { 2 } - 16 } \leq \frac { 2 x } { x ^ { 2 } - 5 x + 4 }$

Match the equation with the appropriate graph. - $f ( x ) = \frac { 18 } { x ^ { 2 } - 9 }$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = ( x + 2 ) ^ { 2 } ( x - 1 )$

Find only the rational zeros. - $f ( x ) = x ^ { 4 } + 25$

Solve the problem. -A stone thrown downward with an initial velocity of $19.6 \mathrm {~m} / \mathrm { sec }$ will travel a distance of $s$ meters, where $\mathrm { s } ( \mathrm { t } ) = 4.9 \mathrm { t } ^ { 2 } + 19.6 \mathrm { t }$ and $\mathrm { t }$ is in seconds. If a stone is thrown downward at $19.6 \mathrm {~m} / \mathrm { sec }$ from a height of $294 \mathrm {~m}$ , how long will it take the stone to hit the ground? Round your answer to the nearest second.

Solve the given inequality (a related function is graphed). - $\frac { 5 x } { x ^ { 2 } - 9 } \geq 0$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $P ( x ) = - 5 x ^ { 4 } + 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 4$

Use the leading-term test to match the function with the correct graph. - $f ( x ) = 2 x ^ { 2 } + 6$

List the critical values of the related function. Then solve the inequality. - $\frac { 1 } { x - 4 } \leq 0$

Find the correct end behavior diagram for the given polynomial function. - $f ( x ) = - x ^ { 5 } - 5 x ^ { 3 } - 2 x + 1$

Match the equation with the appropriate graph. - $f ( x ) = \frac { 18 } { x ^ { 2 } + 9 }$

Classify the polynomial as constant, linear, quadratic, cubic, or quartic, and determine the leading term, the leading coefficient, and the degree of the polynomial. - $f ( x ) = - 13 - x ^ { 2 }$

Solve. -An open-top rectangular box has a square base and it will hold 256 cubic centimeters (cc). Each side of the base has length $x \mathrm {~cm}$ . The box's surface area $S$ is given by $S ( x ) = \frac { 1024 } { x } + x ^ { 2 } .$ Estimate the minimum surface area and the value of $x$ that will yield it.

Use synthetic division to find the function value. - $f ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 5 x - 4 ; \text { find } f ( 5 )$