Math 12 – Chapter 3 Test – Fall ’04     Name______________________________

Write all responses on separate paper.  DO NOT USE A CALCULATOR.

 

1.      Consider the polynomial .  In descending powers,

a.       What sequence of transformations (shifts/shrinks) will transform f to  ?

b.      Apply the leading term test and describe the long-term behavior of this function.

c.       Choose an appropriate value of k so that h(x) satisfies the conditions of the rational zeros theorem, where .  Then, write the possible rational zeros of f(x) which follow from the conclusion of the theorem.

2.      Consider

a.       What does Descartes’ rule of signs say about the number of positive zeros for p? 
What about the number of negative zeros?

b.      Use synthetic division to find the quotient and remainder when  is divided by .

c.       Relate the dividend, divisor, quotient and remainder of part (b) in an equation.

3.      Consider

a.       If x = 2i is a zero, what irreducible quadratic factor does p have?

b.      Find the rational root of p and write p as a product of linear and irreducible quadratic factors.

4.      Let

a.       List all possible rational zeros, according to the theorem on rational zeros.

b.      Use a combination of the rational roots theorem, synthetic division and the theorem on bounds to show that  has no rational roots.

5.      Let .

a.       Use the theorem on bounds to explain why  is an upper bound on the zeros of .

b.      Find all roots of .  Hint :  is a factor.

6.      Sketch a graph of   showing how it passes through all intercepts.

7.      Sketch a graph for the rational function  showing all intercepts and asymptotes.  State what the intercepts and asymptotes are.