Exam 3: Polynomial Functions

Perform the following for the function defined by $f(x)=4 x^{4}-45 x^{2}-49$ . (a) Find all zeros analytically. (b) Find a comprehensive graph of $f$ , and support the real zeros found in part (a). (c) Discuss the symmetry of the graph of $f$ . (d) Use the graph and the results from part (a) to find the solution set of each of the following inequalities. (i) $f(x) \geq 0$ (ii) $f(x)<0$

Perform the following for the function defined by $f(x)=3 x^{4}-11 x^{3}+4 x^{2}+13 x-5$ . (a) List all possible rational zeros. (b) Find all rational zeros. (c) Use the intermediate value theorem to show that there must be a zero between 0 and 1. (d) Use Descartes' rule of signs to determine the possible number of positive zeros and negative zeros. (e) Use the boundedness theorem to show that there is no zero less than -2 and no zero greater than 4 .

The table gives the high school dropout rate (as a percent of enrollment) in the United States for the years 2002 to 2008. (Source: United States Census Bureau, 2008) (a) Plot the data letting $x=2$ represent 2002, $x=3$ represent 2003, and so on. (b) Find a function of the form $f(x)=a(x-h)^{2}+k$ that models this data. Let $(4,4.4)$ represent the vertex, and use (7,3.3) to determine the value of $a$ . (c) Use the statistical capability of a graphing calculator to find the best-fitting quadratic function, $g$ , for this data. Graph both functions $g$ and $f$ from part (b) in the same window as the data points. Which function is the better fit?

Perform the following for the function defined by $f(x)=3 x^{4}-8 x^{3}-6 x^{2}+17 x+6$ . (a) List all possible rational zeros. (b) Find all rational zeros. (c) Use the intermediate value theorem to show that there must be a zero between -2 and -1 . (d) Use Descartes' rule of signs to determine the possible number of positive zeros and negative zeros. (e) Use the boundedness theorem to show that there is no zero less than -2 and no zero greater than 4 .

(a) Solve the quadratic equation $4 x^{2}+2 x-1=0$ analytically. Give the solutions in exact form. (b) Graph $P(x)=4 x^{2}+2 x-1$ with a calculator. Use your results in part (a) along with this graph to give the solution set of each inequality. Express endpoints of the intervals in exact form. (i) $P(x)>0$ (ii) $P(x) \leq 0$

(a) Use only a graphical method to find the real solutions of $x^{5}-3 x^{4}-5 x^{3}+6 x^{2}+2 x+3=0$ . (b) Based on your answer in part (a), how many nonreal (complex) solutions does the equation have?

(a) Use only a graphical method to find the real solutions of $x^{5}-2 x^{4}+2 x^{3}-5 x^{2}+3 x-3=0$ . (b) Based on your answer in part (a), how many nonreal (complex) solutions does the equation have?

For the quadratic function $P(x)=-3 x^{2}-6 x+9$ do each of the following. (a) Find the vertex using an analytic method. (b) Give a comprehensive graph and use a calculator to support your result in part (a). (c) Find the zeros of $P$ and support your result using a graph for one zero and a table for the other. (d) Find the $y$ -intercept analytically. (e) State the domain and range of $P$ . (f) Give the interval over which the function is increasing, and the interval over which it is decreasing.

Perform the following for the function defined by $f(x)=4 x^{4}-5 x^{2}-9$ . (a) Find all zeros analytically. (b) Find a comprehensive graph of $f$ , and support the real zeros found in part (a). (c) Discuss the symmetry of the graph of $f$ . (d) Use the graph and the results from part (a) to find the solution set of each of the following inequalities. (i) $f(x) \geq 0$ (ii) $f(x)<0$

The table gives the number of females who earned doctoral degrees in the United States (in thousands) for the years 1997 to 2006. (Source: Statistical Abstract of the United States, 2006) (a) Plot the data letting $x=7$ represent 1997, $x=8$ represent 1998, and so on. (b) Find a function of the form $f(x)=a(x-h)^{2}+k$ that models this data. Let $(7,19)$ represent the vertex, and use $(16,27)$ to determine the value of $a$ . (c) Use the statistical capability of a graphing calculator to find the best-fitting quadratic function, $g$ , for this data. Graph both functions $g$ and $f$ from part (b) in the same window as the data points. Which function is the better fit?

Perform each operation with complex numbers. Give answers in $a+b i$ form. (a) $(-3-7 i)-(-2-3 i)$ (b) $\frac{10+11 i}{1-4 i}$ (c) Simplify $i^{106}$ (d) $6 i(5-i)^{2}$

The length of a rectangular box is 2 times its height, and its width is 5 more than its height. Find the dimensions of the box if its volume is 792 cubic inches.

(a) Solve the quadratic equation $3 x^{2}+4 x-3=0$ analytically. Give the solutions in exact form. (b) Graph $P(x)=3 x^{2}+4 x-3$ with a calculator. Use your results in part (a) along with this graph to give the solution set of each inequality. Express endpoints of the intervals in exact form. (i) $P(x)>0$ (ii) $P(x) \leq 0$

(a) Use only a graphical method to find the real solutions of $x^{5}-9 x^{4}+4 x^{3}-3 x^{2}+x+2=0$ . (b) Based on your answer in part (a), how many nonreal (complex) solutions does the equation have?

The length of a rectangular box is 10 times its width, and its height is 7 more than its width. Find the dimensions of the box if its volume is 360 cubic inches.

(a) Given that $f(x)=x^{6}-5 x^{5}+3 x^{4}+23 x^{3}-112 x^{2}+162 x-72$ has 1 as a zero of multiplicity $2,-3$ as a single zero, and 4 as a single zero, find all other zeros of $f$ . (b) Use the information from part (a) to sketch the graph of $f$ by hand. Give an end behavior diagram.

Divide. (a) $\frac{x^{3}+3 x^{2}-2 x+2}{x+2}$ (b) $\frac{x^{4}-x^{3}+4 x^{2}-3 x+3}{x^{2}+3}$

(a) Given that $f(x)=x^{6}-9 x^{5}+28 x^{4}-34 x^{3}+40 x-32$ has 2 as a zero of multiplicity 2,4 as a single zero, and -1 as a single zero, find all other zeros of $f$ . (b) Use the information from part (a) to sketch the graph of $f$ by hand. Give an end behavior diagram.

Perform each operation with complex numbers. Give answers in $a+b i$ form. (a) $(-2+4 i)-(3-7 i)$ (b) $\frac{9-7 i}{1-3 i}$ (c) Simplify $i^{57}$ (d) $5 i(3-i)^{2}$

The width of a rectangular box is 4 times its height, and its length is 10 more than its height. Find the dimensions of the box if its volume is 192 cubic inches.