Exam 3: Polynomial and Rational Functions

Solve the problem. -A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow.

Determine whether the function is a polynomial function. - $f ( x ) = 5 x ^ { 7 } - x ^ { 2 } + \frac { 4 } { 3 } x$

Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation. - $x^{3}+9 x^{2}+26 x+24=0$

Determine the constant of variation for the stated condition. -If $y$ varies directly as $x$ , and $y = 300$ when $x = 100$ , find $y$ when $x = 40$ .

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. - $( x + 2 ) ( x - 1 ) ( x - 2 ) < 0$

Determine whether the function is a polynomial function. - $f ( x ) = - 2 x + 6 x ^ { 3 }$

Graph the function. - $f(x)=\frac{x^{2}+4}{x}$

Find the y-intercept of the polynomial function. - $f ( x ) = x ^ { 4 } - 100 x ^ { 2 }$

Find the y-intercept of the polynomial function. - $f ( x ) = x ^ { 3 } + x ^ { 2 } + 1$

Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval notation. - $x < 72 - x ^ { 2 }$

Graph the polynomial function. - $f(x)=6 x^{3}-5 x-x^{5}$

Determine the maximum possible number of turning points for the graph of the function. - $g ( x ) = \frac { 3 } { 2 } x + 1$

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end - $f ( x ) = 4 x ^ { 3 } + 2 x ^ { 2 } - 2 x - 1$

Find the horizontal asymptote, if any, of the graph of the rational function. - $g ( x ) = \frac { 25 x ^ { 2 } } { 5 x ^ { 2 } + 1 }$

The graph of a quadratic function is given. Determine the function's equation. -

Graph the rational function. - $f(x)=\frac{2 x}{x-1}$

Use synthetic division to show that the number given to the right of the equation is a solution of the equation, then solve the polynomial equation. - $3 x ^ { 3 } - 8 x ^ { 2 } - 13 x + 30 = 0 ; 3$

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end - $f ( x ) = - 3 x ^ { 2 } - 3 x - 1$

Solve the problem. -Solve the equation $2 x ^ { 3 } - 17 x ^ { 2 } + 38 x - 15 = 0$ given that 3 is a zero of $f ( x ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 38 x - 15$

Use the Leading Coefficient Test to determine the end behavior of the polynomial function. Then use this end -The following table shows the number of DWI arrests in a county for the years 1994-1998, where 1 represents 1994, 2 represents 1995, and so on. Year, x DWI arrests, T 1994,1 4358.68 1995,2 4429.92 1996,3 4471.5 1997,4 4508 1998,5 4560 This data can be approximated using the third-degree polynomial $T ( x ) = - 0.57 x ^ { 3 } + 0.59 x ^ { 2 } + 63.46 x + 4295.2$ Use this function to predict the number of DWI arrests in 2007. Round to the nearest whole number.