Exam 3: Polynomial Functions

Perform each operation with complex numbers. Give answers in $a+b i$ form. (a) $(8-6 i)-(-7+3 i)$ (b) $\frac{5+14 i}{2+3 i}$ (c) Simplify $i^{76}$ (d) $2 i(3-i)^{2}$

Perform the following for the function defined by $f(x)=2 x^{4}+11 x^{3}+11 x^{2}-18 x-20$ . (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 1 and 2. (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 -6 and no zero greater than 2 .

(a) Use only a graphical method to find the real solutions of $x^{5}+3 x^{4}-5 x^{3}+2 x^{2}-7 x+3=0$ . (b) Based on your answer in part (a), how many nonreal (complex) solutions does the equation have?

Divide. (a) $\frac{x^{3}+2 x+3}{x-2}$ (b) $\frac{2 x^{4}+2 x^{3}-3 x^{2}+3 x-9}{2 x^{2}+3}$

The table gives the average price (in dollars) for a gallon of regular unleaded gasoline as measured in September of the years 2003 to 2009. (Source: The Bureau of Labor Statistics, 2009) (a) Plot the data letting $x=3$ represent 2003, $x=4$ represent 2004, and so on. (b) Find a function of the form $f(x)=a(x-h)^{2}+k$ that models this data. Let $(8,3.70)$ represent the vertex, and use $(3,1.73)$ 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?

The table gives the number of females who earned bachelor's 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 $(7,652)$ represent the vertex, and use $(16,855)$ 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 $f(x)=4 x^{4}+15 x^{2}-4$ . (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 $f(x)=4 x^{4}+7 x^{2}-36$ . (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) $f(x)<0$

For the quadratic function $P(x)=-2 x^{2}-4 x+6$ 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.

For the quadratic function $P(x)=-2 x^{2}+4 x+6$ 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.

Divide. (a) $\frac{x^{3}-3 x^{2}+2}{x-1}$ (b) $\frac{x^{4}-2 x^{3}+2 x^{2}-3}{x^{2}+3}$

Divide. (a) $\frac{6 x^{4}-9 x^{2}}{3 x}$ (b) $\frac{x^{3}-3 x^{2}+5 x-6}{x-1}$

(a) Solve the quadratic equation $2 x^{2}+5 x-1=0$ analytically. Give the solutions in exact form. (b) Graph $P(x)=2 x^{2}+5 x-1$ 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) $P(x) \geq 0$

(a) Given that $f(x)=x^{6}+x^{5}-10 x^{4}+7 x^{3}-x^{2}+10 x-8$ has 1 as a zero of multiplicity 2,2 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.

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 each operation with complex numbers. Give answers in $a+b i$ form. (a) $(2+6 i)-(1-3 i)$ (b) $\frac{7+4 i}{1+2 i}$ (c) Simplify $i^{39}$ (d) $-2 i(5-i)^{3}$

The width of a rectangular box is 3 times its height, and its length is 2 more than its height. Find the dimensions of the box if its volume is 288 cubic inches.

Perform the following for the function defined by $f(x)=2 x^{4}+x^{3}-13 x^{2}-5 x+15$ . (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 3. (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 -3 and no zero greater than 3 .

(a) Solve the quadratic equation $2 x^{2}+5 x-2=0$ analytically. Give the solutions in exact form. (b) Graph $P(x)=2 x^{2}+5 x-2$ 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) Given that $f(x)=x^{6}-2 x^{5}-13 x^{4}+70 x^{3}-160 x^{2}+184 x-80$ has 2 as a zero of multiplicity 2,1 as a single zero, and -5 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.