Name _______________________________________
PC - Take Home #1
70 points
1) Given the function f ( x)  3x 4  x 3 13x 2  5x 10
All work must be shown algebraically for full credit.
a) State the maximum number of real zeros that may exist
[1 point]
b) State the maximum number of turning points
[1 point]
c) Using Descartes’ Rule of Signs determine the possible number of positive and negative
real zeros.
[4 points]
Positive Zeros______________________________________
Negative Zeros_____________________________________
d) Using the Rational Root Theorem to list all the potential rational zeros.
[4 points]
Potential Zeros:_____________________________________________________________
e) Write the polynomial as the product of linear factors.
Do not factor any irreducible factors.
[6 points]
d) List all complex and real roots.
[4 points]
Real Roots_________________________ Complex Roots__________________________
Write the polynomial as the product of linear factors ____________________________________
2)
Let R(x) =
3x 2  3x
x 2  x  12
Find :
a) Domain of R(x)
b) x and y-intercepts, if any, (show algebraically and write your answer as a coordinate)
c) symmetry, if it exists algebraically
d) vertical asymptote(s), if any,
e) horizontal or oblique asymptote, if any
f) Holes, if any
g) Sketch the graph of the rational function. Show ALL asymptotes and intercepts and label the graph.
a)
[3 points]
b)
[3 points]
c)
[2 points]
d)
[4 points]
e)
[2 points]
f)
[1 point]
g) Graph the rational function
[5 points]
Identify each equation. Re- write each equation and put into standard form. If it
is a parabola, give its vertex, focus and directrix, if it is an ellipse, give its center,
vertices, and foci; if it is a hyperbola give its center, vertices, foci, and equations
for asymptotes. Clearly state your answers.
3)
4)
x 2  4 x  8 y  28  0
9 x 2  16 y 2  54x  32 y  79  0
[8 points]
[12 points]
5) Prove the statement for all natural numbers n, using mathematical induction.
Show all steps neatly and clearly for full credit.
2  2 2  2 3  2 4  ...  2 n  2 n1  2
[10 points]