Download 2-13-17 WS Polynomial Applications 2

Algebra II Pre AP
Polynomial Applications Worksheet II
Name ______________________________
Use finite differences to determine if the following data is linear, quadratic, cubic or quartic, then find the
appropriate polynomial function for the given data.
1.
x
f(x)
-3
-1
-2
-7
-1
-3
0
5
1
11
2
9
2.
x
f(x)
-2
-15
-1
-9
0
-9
1
-9
2
-3
Use the calculator to approximate each real zero to the nearest thousandth and find the local maximums and
minimums.
4. f  x   x4  6 x3  3x2  10 x  3
3. f  x   x3  4 x  2
5. Look at the synthetic division chart and answer the following questions:
a) What is the polynomial being divided?
-6
-5
-4
-3
-2
-1
0
1
2
3
b) List the possible rational roots:
c) Make a PNI chart
c) Locate the roots between consecutive integers.
1 5 -3
1 -1 3
1 0 -3
1 1 -7
1 2 -9
1 3 -9
1 4 -7
1 5 -3
1 6 3
1 7 11
1 8 21
d) Are the roots in part c rational, irrational or imaginary? Explain.
e) Based on the synthetic division chart, what is the upper and lower integer bound for the roots?
f) P  3  ________
g) Sketch the graph.
P  1  ________
P  5  ________
-29
2
23
-47 284 -1681
-14 72 -337
-1
6
-1
-2
8
-1
-11 24
-25
-22 24
-1
-29
2
23
-26 -24
-1
-7 -12
-1
34 104
335
Answer the following questions about the stated problem.
6. Given three real numbers, the second number is five more than three times the first number. The sum of all three
numbers is 55.
a) Write a polynomial function for the product of the three numbers in terms of the first number.
b) What three positive numbers, with the relationship described, will maximize the product? (rounded to the nearest thousandth)
c) Find the sets of rational numbers, with the relationship described, whose product is 3066.
7. You are making a rectangular open box out of a 16-inch-by-20-inch piece of cardboard. The box will be formed by
cutting out a square of the same size from each corner and folding up the edges of the piece of cardboard. You want the
box to have the greatest volume possible.
a) Write a polynomial function to represent the volume of the box in terms of its height.
b) What is the greatest volume possible? (rounded to the nearest thousandth)
c) Using the volume function in part (a), find the dimensions of all possible boxes with a volume 100 cubic inches.
8. An open box is to be made from a rectangular piece of sheet metal, 8 inches by 10 inches, by cutting out squares of
equal size from the four corners and bending up the sides. What are the dimensions of the box that would produce the
maximum volume? What is the maximum volume?
Use finite differences to determine if the following data is linear, quadratic, cubic or quartic, then find the
appropriate polynomial function for the given data.
9.
x
-2
f(x) -16
10.
-1
1
0
4
1
5
2
16
11.
x
f(x)
-2
16
-1
7
0
2
1
1
2
4
 1,16  ,  0,9  , 1,10 
 2, 7  ,  3, 12  ,  4, 59 
Solve the following inequalities with a calculator. Express the solution in set notation.
12. x3  4 x  3x 2
13.
 x  7  5x  2 x  62  0
Solve the following inequalities with a calculator. Approximate the intervals to the nearest thousandth, if
necessary. Express the solution in interval notation.
15. 6 x2  9  2 x3
14. 2 x4  5x  16  0
Multiple Choice:
_____ 16. Use a graphing calculator to find the relative minimum, relative maximum, and zeros of
. If necessary, round to the nearest hundredth.
a. relative minimum: (–62.24, 0.36), relative maximum: (37.79, –3.69), zeros: x = 5, –2, 2
b. relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79), zeros: x = –5, –2, 2
c. relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79), zeros: x = 5, –2
d. relative minimum: (–62.24, 0.36), relative maximum: (37.79, –3.69), zeros: x = –5, –2
_____ 17. The volume of a shipping box in cubic feet can be expressed as the polynomial
. Each
dimension of the box can be expressed as a linear expression with integer coefficients. Which expression could represent
one of the three dimensions of the box?
a. x + 6
c. 2x + 3
b. x + 1
d. 2x + 1
_____ 18. The dimensions in inches of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and
height 3x – 1. The volume is about 7.6 ft3. Find the dimensions of the box in inches. Round to the nearest inch.
a. 15 in. by 20 in. by 44 in.
c. 15 in. by 20 in. by 45 in.
b. 12 in. by 17 in. by 35 in.
d. 12 in. by 17 in. by 36 in.
_____ 19. Solve
a. no solution
b. 3, –5
.
c. 3, –3, 5, –5
d. 3, –3
_____ 20. Miguel is designing shipping boxes that are rectangular prisms. One shape of box with height h in feet, has a
volume defined by the function
. Graph the function. What is the maximum volume for the
domain
? Round to the nearest cubic foot.
a. 10 ft3
b. 107 ft3
c. 105 ft3
d. 110 ft3
2
-7
2
14
-12
21. Use the synthetic division table to answer the following:
a) Make a PNI chart.
-4
2
-15
62
-234
924
-3
2
-13
41
-109
315
b) List the possible rational roots.
-2
2
-11
24
-34
56
-1
2
-9
11
3
-15
0
2
-7
2
14
-12
1
2
-5
-3
11
-1
d) Based on the synthetic division chart, what is the upper and
2
2
-3
-4
6
0
lower integer bound for the roots?
3
2
-1
-1
11
21
e) From your list in part b, what are the only numbers that need
4
2
1
6
38
140
c) How many real roots are there?
c) Locate roots between consecutive integers.
to be considered possible roots?
f) Find all roots.