## Math 1150: Precalculus, Instructor: Dana Ernst

### Review for Midterm Exam  #1

Here is a review for your upcoming exam.  The second exam covers all of Chapter 2 (sections 2.1-2.7).  I will not collect this.  The test that I give you will be very similar to this review; however, I will not limit myself to questions identical to these.  This review will give you a good indication of what you will be expected to know for the exam.  Obviously, this review is longer than the exam will be.  In a couple of days, I will provide you with a list of answers for each of the following problems.

Reminders:  During the exam, SHOW ALL YOUR WORK and JUSTIFY YOUR ANSWERS when necessary.  Also, be organized and neat!

Note:  On any problem where you need to use the Rational Zero Theorem, you should use a calculator to limit the number of possibilities.  During the exam, you will not be allowed to use a calculator.  Instead, I will either design the problem so that there are not many possibilities or I will provide you will some extra information.

1. Simplify each of the following expressions.  Write each of your answers in standard form.

(a)

(b)

(c)

(d)

(e)   .

2.     Solve each of the following inequalities and graph the solutions on the real number line.

(a)

(b)

(c)

(d)

3.     Sketch the graph of  without using your calculator.  Be sure to plot any x or y-intercepts.

4.     Write the standard form of the equation of the parabola with vertex at  and passing through the point .

5.     Identify the vertex of .

6.     Determine the minimum value of .

7.     Exercise #113, page 178.

8.     Exercise #81, page 115.

9.     What is the maximum number of zeros a polynomial of degree n can have?

10.  Is it possible for a polynomial of degree 6 to have exactly 5 distinct real roots?  If it is possible, give an example.  If not, explain why.

11.  Is it possible for a polynomial of degree 5 to have 2 real roots and 3 complex roots? If it is possible, give an example.  If not, explain why.

12.   What is the maximum number of turning points a polynomial of degree n can have?

13.  Find all the real zeros of .

14.  Find a polynomial function that has the following zeros: 0, -1, 2.

15.  Sketch the graph of  without using your calculator.  Be sure to plot any x or y-intercepts.

16.  Use long division to divide the following expression: .

17.  Determine which of , 1, and 4 is a solution to .

18.  Explain the relationship between zeros of a polynomial and the factors of  a polynomial.

19.  Using the fact that 2 is a zero of , factor the function completely.

20.   List all of the possible rational zeros of .

21.  Find all of the rational zeros of .

22.  Find all the real zeros of .

23.  Write  in completely factored form.  Include any complex number factors.

24.  Use the fact that  is a zero of  to find any remaining zeros.

25.  Find all the zeros of .

26.  Find any horizontal and vertical asymptotes of .

27.  Sketch the graph of each of the following functions without using your calculator. Be sure to find any asymptotes and x and y-intercepts.  Also, state domain.

(a)     (Donâ€™t bother to find the x-intercepts for this one)

(b)

(c)

28.  Follow the directions for each of the following.

(a)   Exercise #13, page 146

(b)  Exercise #21, page 146

(c)   Exercise #35, page 147

(d)  Exercise #41, page 147

(e)   Exercise #43, page 147

(f)   Exercise #47, page 147

(g)   Exercise #63, page 147

(h)  Exercise #73, page 147