1.  Find a fifth degree polynomial that will have the zeros 0, 1, -2 and i.

a.x5  + x4  -  x3  +  x2  -  2x
b.x5  - x4  -  x3  +  x2  -  2x
c.x4  + x3  -  2x2  -  ix3  -  ix2  +2ix
d.x4  - x3  -  2x2  -  ix3  +  ix2  +2ix
e.none of these

2. Find a fourth degree polynomial that has zeros –1 (with a multiplicity of 2) and 1 – i.

a.x2 – 2x – ix + 1 – i 
b.x2 + 2x – ix + 1 – i 
c.x4 – 4x3 + 7x2 – 4x + 2
d.x4 + 2x3 + x2 + 2ix2 + 4ix + 2i
e.none of these

Questions 3 – 12 refer to the polynomial  
f(x)  =  x5  +  x4  -  15x3  -  x2  +  38x  -  24.

3. How many positive real zeros MIGHT this polynomial have?

a.4 or 2 or 0
b.3 or 1
c.4 or 2
d.3 or 1 or 0
e.none of these

4. How many negative real zeros MIGHT this polynomial have?

a.4 or 2 or 0
b.3 or 1
c.4 or 2
d.3 or 1 or 0
e.none of these

5.  What are the POSSIBLE rational zeros?

a.+- 2, +- 3, +- 4, +- 6, +- 8, +- 12
b.1, 2, 3, 4, 6, 8, 12,  24
c.+- 1, +- 2, +- 3, +- 4,  +- 6, +- 8, +- 12, +- 24
d.+- 1, +- 24
e.none of these

6.  Use synthetic division to identify all  rational zeros.

a.1, 2, 3, 4
b.–4, -2, 1, 3
c.–4, -3, -2, -1
d.–3, -1, 2, 4
e.none of these



7. Are there any complex zeros? If so, what are they?

a.yes, i and – i
b.yes, 2i and –2i
c.yes, i and –2i
d.yes, -i and 2i
e.none of these

8.  Factor the polynomial completely.

a.(x – 1)(x – 2)(x – 3)(x – 4)
b.(x + 1)(x + 2)(x + 3)(x + 4)
c.(x+1)2(x – 2)(x + 3)(x – 4)
d.(x – 1)2(x + 2)(x – 3)(x + 4)
e.none of these

9.  What are the x-intercepts of the graph of this polynomial?

a.1, 2, 3, 4
b.–4, -2, 1, 3
c.–4, -3, -2, -1
d.–3, -1, 2, 4
e.none of these

10.  What is the multiplicity of the smallest zero, and what does this say about the graph?

i.multiplicity is 1
ii.multiplicity is 2
iii.multiplicity is 3
iv.graph “touches” the x-axis at this intercept
v.graph “goes through” the x-axis at this intercept.
a.i and iv
b.i and v
c.ii and iv
d.ii and v
e.iii and iv
f.iii and v
g.none of these

11.  What is the y-intercept of this function (where does the graph cross the y-axis)?

a.1
b.0
c.–24
d.24
e.none of these

12.  What is the end-behavior of the graph (does it rise or fall at each end)?

a.rises at both ends
b.falls at both ends
c.rises at left end, falls at right end
d.falls at left end, rises at right end
e.none of these


13.  The answer to this question is E.  
Questions 14 – 23 refer to the polynomial
f(x) = x4 – 4x3 + 7x2 – 16x + 12.

14. How many positive real zeros MIGHT this polynomial have?

a.4 or 2 or 0
b.3 or 1
c.4 or 2
d.3 or 1 or 0
e.none of these

15. How many negative real zeros MIGHT this polynomial have?

a.4 or 2 or 0
b.3 or 1
c.4 or 2
d.3 or 1 or 0
e.none of these

16.  What are the POSSIBLE rational zeros?

a. 2,  3,  4,  6
b.1, 2, 3, 4, 6, 12
c. 1,  2,  3,   4  6,   12
d. 1,  12
e.none of these

17.  Use synthetic division to identify all  rational zeros.

a.1, 3
b.-1, -3
c.1, -3
d.–1, 3
e.none of these

18. Are there any complex zeros? If so, what are they?

a.  yes, i and – i
b.yes, 2i and –2i
c.yes, i and –2i
d.yes, -i and 2i
e.none of these

19.  Factor the polynomial completely.

a.(x – 1)(x – 3)(x – 2i)(x + 2i)
b.(x + 1)(x + 3)(x – 2i)(x + 2i)
c.(x – 1)(x – 3)(x – i)(x + i)
d.(x + 1)(x + 3)(x – i)(x + i)
e.none of these

20.  What are the x-intercepts of the graph of this polynomial?

a.1, 3
b.-1, -3
c.1, -3
d.–1, 3
e.none of these


21.  What is the multiplicity of the smallest zero, and what does this say about the graph?

i.multiplicity is 1
ii.    multiplicity is 2
iii.   multiplicity is 3
iv.   graph “touches” the x-axis at this intercept
v.   graph “goes through” the x-axis at this intercept.
a.i and iv
b.i and v
c.ii and iv
d.ii and v
e.iii and iv
f.iii and v
g.none of these

22.  What is the y-intercept of this function (where does the graph cross the y-axis)?

a.0
b.1
c.12
d.–12
e.none of these

23.  What is the end-behavior of the graph (does it rise or fall at each end)?

a.rises at both ends
b.falls at both ends
c.rises at left end, falls at right end
d.falls at left end, rises at right end
e.none of these

24.  Solve the system

If the solution point is  (x , y), what is the product xy? (ie, multiply the coordinates of the solution point)

a.0
b.2
c.–1
d.–12
e.none of these

25.  Solve the system

If the solution point is  (x , y), what is the product xy? (ie, multiply the coordinates of the solution point)

a.0
b.2
c.–1
d.–12
e.none of these