M121 Fall 95  Exam I  Sept. 15, 1995

Name (Print) SSN: Circle the name of your TA and Discussion Hour:

David Krebes  8 AM  11 AM
Qihui Liu:  8am  9am
Jukun Ying  9am  11 am

M121 Fall 95  Exam I  Sept. 15, 1995

Answer all questions and SHOW ALL WORK. No credit for answers only.

Part I: Multiple choice. No partial credit but SHOW WORK! Circle the letter corresponding to the correct answer. The next two questions refer to the polynomial:

f(x) = 5x^{3} + 2x^{2} + −5x + 2.

1. Use the rational roots theorem to list all possible candidates for rational zeroes of the polynomial f(x).

(A) 1, 2, ±5 (B) ±1, ±2, ±1/5, ±2/5 (C) ±1, ±1/2, ±5, ±5/2 (D) ±1, ±1/2, ±5, ±2/5 (E) ±1, ±2, ±5

2. (a) Use a graph to eliminate some or all of the candidates from the list in the previous question. Copy a sketch of the graph from your calculator and indicate which roots have been eliminated.

(b) Find all real zeros of f(x).

3. Which of the following is a polynomial with real coefficients that has 2 and 2 − i as zeros? (A) (x + 2)(x − 2 − i) (B) (x − 2)(x + 2 + i) (C) (x − 2)(x^{2} − 4x + 5) (D) (x + 2)(x^{2} − 4x + 5) (E) (x + 2)(x^{2} + 5)

M121 Fall 95  Exam I  Sept. 15, 1995

4. The figure below shows a rectangle with its base on the x-axis and its two upper vertices on the graph of the equation y = 8 − x^{2}. Which of the following represents the area of this rectangle as a function of x.

(A) f(x) = 16x − 2x^{3} (B) f(x) = 16 − 2x^{2} (C) f(x) = 8 − x^{2} (D) f(x) = 8x − x^{3} (E) none of the above

5. The graph of f(x) = p(x)/q(x) where p and q are polynomials are is sketched below. Which of the following descriptions of p and q may be correct?

(A) The degree of q is greater than the degree of p. (B) The degree of q is one and the degree of p is three. (C) q has no real zeros and the degree of p is two greater than the degree of q. (D) p and q have the same degree. (E) The degree of q is one and the degree of p is two.

M121 Fall 95  Exam I  Sept. 15, 1995

6. The graph of y = −3 + \frac{4}{x+2} can be obtained from the graph of y = 1/x by applying in

order the following three transformations.

1. vertical stretch by a factor of a

2. horizontal shift of b units left or right 3. vertical shift of c units up or down What are the values of a, b, c and in which directions are the shifts?

(A) a =2, b=3 left, c =4 up (B) a =2, b=3 right, c =4 up (C) a =2, b= 2 right, c =3 down (D) a =2, b= 2 right, c =3 up (E) a =2, b= 2 left, c =3 down

(Partial credit) Write complete solutions and indicate your answers clearly.

7. Find exactly all real zeros of the polynomial f(x) = 2x^{3} + 14x + 20x. For which x is f(x) > 0?

8. The sketch below is of one of the following five functions, which one? (A) (x^{2} − 5)/(x + 1)(x^{2} + 1) (B) (x^{2} + 5)/(x + 1)(x^{2} + 1) (C) (x^{2} − 5)/(x − 1)(x^{2} + 1) (D) (x^{2} − 5)/(x + 1)(x^{2} + 1) (E) (x^{2} + 5)/(x − 1)(x^{2} + 1))

M121 Fall 95  Exam I  Sept. 15, 1995

Part II : (Partial credit) Write complete solutions and indicate your answers clearly.

9. Consider the function f(x) = x^{4} − x^{2} − 1. Sketch the complete graph of f(x). Half the credit on this problem will be awarded for your argument that this graph is complete. Explain why your sketch identifies all the zero’s of f.