Exam 4: Polynomial and Rational Functions

The graph of the function $f ( x ) = \frac { 3 x } { x + 1 }$ is shown below. Determine the vertical and horizontal asymptotes of its graph.

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } + 28 x ^ { 2 } + 25 x + 6$ .

Find all the real zeros of the polynomial function $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$ and determine the multiplicity of each.

Describe the right-hand and the left-hand behavior of the graph of $q ( x ) = - 3 x ^ { 4 } + 7 x ^ { 3 } + 27$ .

Use long division to divide. $\left( 2 x ^ { 2 } + 7 x + 3 \right) \div ( x + 3 )$

Simplify and write the following complex number in standard form. $\sqrt { - 684 } \cdot \sqrt { - 19 }$

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + x - 2 }$ .

Find the real zeros of the polynomial function $f ( x ) = x ^ { 5 } - 16 x ^ { 4 } + 64 x ^ { 3 }$ and determine the multiplicity of each.

Determine the vertex of the graph of the quadratic function $f ( x ) = x ^ { 2 } + x + \frac { 5 } { 4 }$ .

Determine the equations of the vertical and horizontal asymptotes of the graph of the function $f ( x ) = \frac { x } { x - 3 }$ .

Determine which of the following rational functions has a vertical asymptote of $x = - 2 , x = - 3$ and no horizontal asymptote. Function I : $f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } + 5 x + 6 }$ Function IV : $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - x - 6 }$ Function II : $f ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 5 x + 6 }$ Function V : $f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - x - 6 }$ Function III : $f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 5 x + 6 }$

Find an equation of the parabola below.

Describe the right-hand and the left-hand behavior of the graph of $u ( x ) = - \frac { 3 } { 7 } \left( x ^ { 3 } + 10 x ^ { 2 } - 3 x + 1 \right)$ .

Write $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x - 29$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when $k = 6$ .

Find a polynomial function with real coefficients that the given zeros. $- 5 , - 5,1 + \sqrt { 3 } i$

Write $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 25 x - 75$ as a product of linear factors.

Find all the zeros of the polynomial function $P ( x ) = x ^ { 3 } + 10 x ^ { 2 } + 34 x + 40$ and write the polynomial as a product of its linear factors.

Using the factors $( x - 5 )$ and $( x + 3 )$ , find the remaining factor(s) of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15$ and write the polynomial in fully factored form.

Use synthetic division to divide. $\left( x ^ { 3 } - 48 x + 128 \right) \div ( x - 4 )$

Examine the leading term and determine the far-left and far-right behavior of the graph of the polynomial function $P ( x ) = 5 x ^ { 5 } + 5 x ^ { 3 } - 2 x ^ { 2 } + 8$ .