Exam 3: Polynomial and Rational Functions

Find all zeros of the polynomial $P ( x ) = x ^ { 5 } + x ^ { 3 }$ .

Find the maximum or minimum value of the function $f ( x ) = 3 - 8 x - 2 x ^ { 2 }$ .

Sketch the graph of the function $p ( x ) = - ( x + 5 ) ^ { 3 } + 8$ by transforming the graph of an appropriate function of the form $y = x ^ { n }$ . Indicate all $x$ - and $y$ - intercepts and make sure your graph exhibits the proper end behavior.

Find all rational zeros of the polynomial $p ( x ) = x ^ { 4 } - 9 x ^ { 3 } + 15 x ^ { 2 } - 7 x$ .

Find all rational zeros of the polynomial $p ( x ) = 5 x ^ { 5 } - x ^ { 4 } - 10 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 1$ .

Find the quotient and remainder of $\frac { x ^ { 3 } + 3 x - 2 } { x - 1 }$ .

Find a polynomial $P$ of degree $3$ , with integer coefficients that has zeros of $1 - \sqrt { 3 } i$ and $0$ .

Find the polynomial of degree $3$ that has zeros $- 2$ , $- 1$ , and $3$ .  

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

Sketch the graph of the parabola $y = \frac { 1 } { 3 } x ^ { 2 } - \frac { 2 } { 3 } x$ and state the coordinates of its vertex and its intercepts.

Give an example of a polynomial that has $6$ local extrema.

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

Evaluate the expression and write the result in the form a + bi. $\frac { 36 } { 6 - 6 i }$

Find all solutions of the equation $4 x ^ { 2 } - 7 x = - x - 22$ and express them in standard form $a + b i$ .

Sketch the graph of the parabola $y = - 2 x ^ { 2 } - 19 x + 10$ and state the coordinates of its vertex and its intercepts.

Find the remainder of $p ( x ) = 22 x ^ { 2004 } - 5 x ^ { 569 } + x ^ { 104 } - 27 x + 22 x ^ { 2003 } - 5 x ^ { 568 } + x ^ { 103 } - 27$ , if $p ( x )$ is divided by $( x + 1 )$ . What is the significance of this remainder?

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

Factor the polynomial $p ( x ) = x ^ { 2 } + 4$ completely and find all its zeros. State the multiplicity of each zero.

Find a polynomial $P$ of degree $4$ , with integer coefficients that has zeros of $- 4 i$ , and $5 i$ , and leading coefficient $3$ .

Evaluate the expression $\frac { 4 } { 3 - \sqrt { - 4 } } + \frac { i } { 3 + \sqrt { - 4 } }$ and write the result in the form $a + b i$ .