College-Algebra, Take Home Test 2
Chapter 3: Polynomial Functions
Name_______________________
Print, solve all problems completely, and bring it on the day of the 2nd on campus exam for full
credit.
1. Evaluate the expression :
−12 ⋅ −6
a.
8
b.
2 + −9
−4 − −32
ai 1000
2. Evaluate the expression :
ai −81
ai 103
3. Evaluate the expression : 2 + 3i
1 − 5i
4. Determine the number of real solutions. a) 2x 2 = 5x − 3
bx 2 + x − 4 = 0
5. Find the remainder of : x 4 + 3x − 8 ÷ x + 1
6. Find pk for :px = x 2 + 3x − 10 when k = −3
7. Is −4 a zero of the polynomial px = 6x 3 + 25x 2 + 3x − 4? Justify your answer.
8. Find a polynomial function (with real coefficient and of least possible degree)having the
following zeros: 3 , − 3 , 2, 3
9. If 2i is a zero of px = 2x 4 − x 3 + 7x 2 − 4x − 4,
a. Find all linear factors of p(x).
b. If p(x) is divided by x + 4 , What is the remainder?
10. Given the graph of fx : x 2 + x − 2
y 25
20
15
10
5
-5
a. Solve fx = 0
b. Solve fx < 0
c. Solve fx > 0
1
-4
-3
-2
-1
1
2
3
4
5
x
11. The function defined by fx = 5x 3 − 2x 2 − 5 has ____ complex zeros and at least ______real
zeros.
12. The function defined by fx = −3x 4 + 5 has ____ complex zeros and has ______real zeros.
13. Does px = 3x 2 − 2x − 6 have a zero between 1 and 2? Why or why not?
a. Graph it using a graphing calculator.
b. What are the coordinates of the extreme point of the graph.
c. Give the intervals of increasing and decreasing.
d. Determine the maximum or minimum value of the function.
14. Give all values of k that ensures exactly one real solution for kx 2 + 16x + k = 0
15. Given.gx = 2x 2 − 2x − 24
a. Express the quadratic function in standard form. gx = ax − h 2 + k
b. Find all the intercepts.
c. Find its maximum or minimum value.
d. Give the axis of symmetry.
e. Sketch its graph.
16. Find the equation of the quadratic function satisfying the given
conditions. px = ax − h 2 + k Express your answer in the form px = ax 2 + bx + c.
vertex:−2, −3; through: 0, −19
17. Solve: x 2 + 6x ≥ −8
18. A farmer has 80 feet of fence to enclose a rectangular garden. If the length must be twice the
width, find the dimensions that will maximize the area enclosed.
19. Sketch and justify (using the end behavior and the behavior near an x-intercept & the
y-intercept) the graph of::
a. fx = x + 4 2 x − 1.
b. fx = x + 3x + 1 2 x − 1.
20. Find a polynomial with integer coefficients that satisfies the given conditions.
fx has degree 5 zeros 4, −2, and 1 + 2i, and leading coefficient −3; the zero −2 has multiplicity
2.
21. Find all the zeros for px = x 3 − 5x 2 + 17x − 13 if we know 1 is a zero.
22. Give the equation
2x 3 − 3x 2 − 8x + 12 = 0
a. Find all possible rational zeros.
b. Prove that 3 is an upper bound.
c. Prove that −3 is a lower bound.
2
d. How many positive, negative and imaginary roots can the equation have?
23. Give the equation
2x 4 − 5x 3 + 3x + 1 = 0
a. Find all possible rational zeros.
b. How many positive, negative and imaginary roots can the equation have?
3