Exam 2: Polynomial and Rational Functions

The graph of the function $f ( x ) = \frac { 3 x } { x + 1 }$ is shown below. Determine the domain.

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + 2$ .

Write the polynomial in completely factored form. (Hint: One factor is $x ^ { 2 } + 1 .$ ) $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 14 x ^ { 2 } + 6 x + 13$

Simplify $( - 4 + i ) ( 4 - 5 i )$ and write the answer in standard form.

If $x = 4$ is a root of $x ^ { 3 } + 5 x ^ { 2 } - 16 x - 80 = 0$ , use synthetic division to factor the polynomial completely and list all real solutions of the equation.

Which of the following is the graph of the given equation? $f ( x ) = \frac { 2 - x } { x + 3 }$

Use long division to divide. $\left( x ^ { 4 } - x ^ { 2 } - 5 \right) \div \left( x ^ { 2 } + 4 x - 1 \right)$

Use synthetic division to divide. $\left( 2 x ^ { 3 } - 5 x ^ { 2 } - 22 x - 15 \right) \div ( x - 5 )$

Find all real zeros of the polynomial $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 49 x - 147$ and determine the mutiplicity of each.

Sketch the graph of the rational function below. $f ( x ) = \frac { x ^ { 2 } - 1 } { x }$

Determine the domain of the function $f ( x ) = \frac { x ^ { 2 } + 7 x + 12 } { x ^ { 2 } + 16 }$ .

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

Which of the given graphs is the graph of the polynomial function below? $h ( x ) = x ^ { 5 } - \frac { 3 } { 2 } x ^ { 3 } - \frac { 1 } { 2 } x$ Graph 1: Graph 2 Graph 3 : Graph 4 : Graph 5 :

Determine the zeros (if any) of the rational function $g ( x ) = 5 + \frac { 2 } { x ^ { 2 } + 5 }$ .

Find all zeros of the function $f ( x ) = x ^ { 2 } ( x - 2 ) \left( x ^ { 3 } - 216 \right)$ .

Find the zeros of the function below algebraically, if any exist.

Write the standard form of the equation of the parabola that has a vertex at $( 3,8 )$ and passes through the point $( 5 , - 2 )$ .

Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$ . Determine the domain of $f ( x )$ and find any vertical asymptotes.

Find the quadratic function $f$ whose graph intersects the $x$ -axis at $( 1,0 )$ and $( 4,0 )$ and the $y$ -axis at $( 0,4 )$ .

Compare the graph of $p ( x ) = \left[ - \frac { 1 } { 3 } ( x + 9 ) \right] ^ { 2 } - 9$ with $p ( x ) = x ^ { 2 }$ .