Exam 3: Polynomial and Rational Functions

Find all rational zeros of the polynomial $p ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 17 x - 60$ .

Sketch the graph of the function $p ( x ) = - x ^ { 4 } + 8 x ^ { 3 } - 16 x ^ { 2 }$ . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Sketch the graph of the parabola $y = - x ^ { 2 }$ and state the coordinates of its vertex and its intercepts.

Graph the rational function $f ( x ) = \frac { x ^ { 4 } - 3 x ^ { 3 } + x ^ { 2 } - x - 12 } { 2 x ^ { 3 } + 5 x ^ { 2 } - 28 x - 15 }$ and determine all vertical asymptotes. Then graph $f$ and $g ( x ) = \frac { 1 } { 2 } x - \frac { 11 } { 4 }$ , on the same screen, to show that they have the same end behavior.

Find the complete factorization of $P ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 1$

Use a graphing device to find all real solutions of the equation $3 x ^ { 4 } + 8 x ^ { 3 } + 2 x ^ { 2 } - 5 x - 3 = 0$ .

An open box with volume of $8000$ cm $3$ is to be constructed by taking a piece of cardboard $40$ cm by $60$ cm, cutting squares of side length $x$ cm from each corner. Show that this can be done in two ways, and find the exact dimensions of the box in each case.

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

Among all rectangles that have a perimeter of $24$ ft, find the dimensions of the one with the largest area.

Factor the polynomial $p ( x ) = x ^ { 8 } + 16 x ^ { 5 } + 64 x ^ { 2 }$ completely and find all its zeros. State the multiplicity of each zero.

Sketch the graph of the function $p ( x ) = - \frac { 1 } { 2 } x ^ { 5 } + x ^ { 4 } + 2 x ^ { 3 } - 4 x ^ { 2 }$ . Make sure your graph shows all intercepts and exhibits the proper end behavior.

Find the maximum or minimum value of the function $f ( x ) = 7 + 12 x - 3 x ^ { 2 }$ .

Sketch the graph of the function $y = ( x - 3 ) ^ { 2 }$ 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 rational zeros of the polynomial $p ( x ) = 2 x ^ { 6 } - 11 x ^ { 5 } + 27 x ^ { 4 } - 57 x ^ { 3 } + 79 x ^ { 2 } - 52 x + 12$ .

Let $p ( x ) = - x ^ { 3 } - 3 x ^ { 2 } + 33 x + 35$ . Show that $p ( - 1 ) = 0$ , and use this fact to find all the zeros of $p ( x )$ .  

Use synthetic division and the remainder theorem to evaluate $p \left( \frac { 1 } { 2 } \right)$ for $p ( x ) = 2 x ^ { 2 } - 2 x + 2$ .

Let $f ( x ) = x ^ { 3 } + c$ . Graph the family of functions with $c = - 4$ , $- 2$ , $0$ , $2$ , and $4$ in the viewing rectangle $[ - 5,5 ]$ by $[ - 5,5 ]$ . How does the value of $c$ affect the graph?

Evaluate the expression and write the result in the form a + bi. $\left( 7 - \frac { 5 } { 2 } i \right) - \left( 5 + \frac { 7 } { 2 } i \right)$

Use the Factor Theorem to show that $( 2 - x )$ is a factor of the polynomial $P ( x ) = x ^ { 3 } - 13 x ^ { 4 } + 57 x ^ { 2 } - 83 x ^ { 2 } - 34 x + 120$ .

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). $f ( x ) = 2 x ^ { 2 } + 8 x + 9$