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# Different algebra problems

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1. The result of dividing x4 - 2x3 + x2 - 3x + 2 by x - 2 is:
A. x3 - x + 1
B. x3 - x - 1
C. x3 + x - 1
D. x3 + x + 1

2. What is the remainder when 4x3 + 5x2 - 13x - 7 is divided by x - 2?
A. -33
B. 19
C. 33
D. 7

Use the Remainder Theorem to find P(c). P(x) = x4 + 3x2 + 1, c = 3
A. 107
B. 109
C. 19
D. 1

4. Are (x - 1) and (x + 4) factors of P(x) = 2x3 + 5x2 - 13x + 4 ?
A. Neither
B. Both
C. Only (x - 1)
D. Only (x + 4)

5. Determine whether -3 and 2 are zeros of P(x) = 2x3 - 5x2 - 4x + 12.
A. Neither
B. Both
C. Only -3
D. Only 2

6. What is the remainder if 7x53 - 3x31 + 19x3 - 13 is divided by x + 1?
A. 10
B. -42
C. -36
D. -13

7. Determine the far-left and far-right behavior of the graph of the polynomial function P(x) = 3x4 - 5x2 + 7.
A. up to the left and down to the right
B. down to the left and down to the right
C. up to the left and up to the right
D. down to the left and up to the right

8. Determine the maximum or minimum range value [P(x)] of the function P(x) = -3x2 + 6x + 5. State whether this value is a maximum or a minimum.
A. 1; minimum
B. 8; maximum
C. 1; maximum
D. 8; minimum

9. Sketch the graph of P(x) = 4x2 - x3.

10. Find the real zeros of the polynomial function P(x) = x(x + 1)(x2 - 81).
A. 0, -1, 9, -9
B. 0, -1, 9
C. -1, 9, -9
D. 0, 1, 9

11. Find all the real numbers that solve: x(x2 + 64) = 0.
A. -8, 0, 8
B. 0, 8
C. -8, 0
D. 0

12. Find all the factors of x3 + 2x2 - 3x.
A. (x-3)(x+1)
B. x(x-3)(x+1)
C. x(x+3)(x-1)
D. (x+3)(x-1)

13. A manufacturer has determined that the revenue received from selling x items of a product is given by R(x) = -0.3x2 + 200x and the cost to produce x items of the product is given by C(x) = 80x + 9400. Assuming that all of the products produced can be sold, how many should be produced to maximize the profit? (Hint: profit = revenue - cost)
A. 0.00125
B. 400
C. 200
D. 33

14. Use the Rational Zero Theorem to list the possible rational zeros of the polynomial 4x4 - 9x2 + 5.

15. Using Descartes Rule of Signs, determine the number of possible positive real roots of the polynomial 3x4 + 5x3 + 8x2 - 7x + 2.
A. 4, 2 or 0
B. 2 or 0
C. 2
D. none

16. With the help of synthetic division, find the zeros of the polynomial 2x3 + 9x2 - 32x + 21.
A. -1/7, 1, 2/3
B. -3/2, -1, 7
C. -7, 1, 3/2
D. -7, -3/2, 1

17. Find all the zeros of the polynomial x3 - 4x2 + 9x - 36.
A. -4, 3i, -3i
B. -3, 4i, -4i
C. 4, 3i, -3i
D. 3, 4i, -4i

18. Write x4 - 6x2 - 27 as a product of linear factors and quadratic factors that are irreducible over the reals.
A. (x2 + 3)(x + 3)(x - 3)
B. (x + 3)2(x - 3)
C. (x2 + 3)(x2 + 9)
D. (x + 3)2(x - 3)2

19. Find a polynomial of degree 3 that has zeros 3, 2, and -1.
A. x3 - 7x - 6
B. x3 - 4x2 + x + 6
C. x3 + 4x2 + x - 6
D. x3 - 6x2 - x + 6

20. Two hundred meters of rope is used to mark off a rectangle (the dimensions are x meters by y meters.
a. Write the equation showing perimeter of 200 using x and y.
b. Write the formula for area using x and y.
c. Write the function for area using only x, which could be used to find the maximum possible area

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## Practice Problem - Your group will develop four different population scenarios for a town. As a group, you will decide on the name of the town and the initial population. You will graph the function for each population scenario and use your model to make some decisions about the population. ....[See the attached questions file.]

Your group will develop four different population scenarios for a town. As a group, you will decide on the name of the town and the initial population. You will graph the function for each population scenario and use your model to make some decisions about the population.
1) Decide on a name of a rural town. Name of town: _______________
2) Decide on an initial population, , of the town in the year 2010. Choose an initial population between 1000-5000. Use this value of for each of the scenarios.

P0 = ___________

3) You will investigate four different scenarios of population growth or decline in this town.
? Linear growth
? Growth modeled by a quadratic equation
? Growth modeled by a radical equation
? Population decline modeled by a rational equation
I. Linear Growth:
Suppose that the amount that your town's population grows each year is fixed (or constant).

Choose the amount of population growth each year = _______
(Hint: Choose a whole number for your growth rate, rather than a percent.)

a) Fill in the following chart:

Year (t) Population (P)
t = 0
(2010) ______
t = 1
(2011)
t = 2
(2012)
t = 3
(2013)
t = 6
(2016)
b) Find a linear equation in the form P = mt + b (y = mx + b), which gives the population, P, t years from 2010.

Solution:

c) Use your equation in part b to approximate the population in the year 2020.

Solution

d) Use your equation in part b to approximate how many years it will take the population to reach 12,000. Round to the nearest whole year when necessary.

Solution

e) Graph this function in MS Excel by plotting the points found in your chart in part a. You may also use another web-based graphing utility. Label your axes with time on the x-axis and population on the y-axis. Copy and paste your graph here:

where t is the time in years from 2010.

a) Insert the value of that your group has decided upon into the equation. Use t^2 to type t-squared.

b) Fill in the following chart.

Year (t) Population (P)
t = 0
(2010) _______
t = 1
(2011)
t = 2
(2012)
t = 3
(2013)
t = 6
(2016)

c) Use your equation from part a to approximate how many years it will take for the population to reach 12,000. Round to the nearest whole year when necessary.
Solution

d) Graph this function in MS Excel by plotting the points found in your chart in part a. You may also use another web-based graphing utility. Label your axes with time on the x-axis and population on the y-axis. Copy and paste your graph here:

III. Growth Modeled by a Radical Equation:

Suppose instead that the town experiences growth that can be modeled by the following: where t is the number of years from 2010.

a) Insert the value of that your group has decided upon into the equation above. Use the Equation Editor or type square root of t as sqrt(t).

b) Fill in the following chart. Round to the nearest whole person when necessary.

Year (t) Population (P)
t = 0
(2010) _______
t = 1
(2011)
t = 2
(2012)
t = 3
(2013)
t = 6
(2016)

c) Use your equation from part a) to approximate how many years it would take for the population to reach 12,000. Round the nearest whole year when necessary.

d) Graph this function in MS Excel by plotting the points found in your chart in part a. You may also use another web-based graphing utility. Label your axes with time on the x-axis and population on the y-axis. Copy and paste your graph here:

IV. Population Decline Modeled by a Rational Equation:

Suppose instead that the town experiences population decline that can be modeled by the following: where t is the number of years from 2010.

a) Insert the value of that your group has agreed to use.
Type as ( ) / (t + 1) or use the Equation Editor.

b) Fill in the following chart. Round to the nearest whole person when necessary.

Year (t) Population (P)
t = 0
(2010) ________
t = 1
(2011)
t = 2
(2012)
t = 3
(2013)
t = 6
(2016)

c) Use your equation from part a) to approximate how many years it would take for the population to reach 400. Round to two decimal places if necessary.

Solution

d) Graph this function in MS Excel by plotting the points in the chart in part a. You may also use another web-based graphing utility. Label your axes with time on the x-axis and population on the y-axis. Copy and paste graph here:

V.
Suppose that the mayor of the town you have chosen has built a new factory in hopes of drawing as many new people to the town as possible. Which one of the four models would the mayor hope that the population would follow? Explain.

6) State the domain of the following:

a)

b)

c)

d)

e)

7) Suppose the graph of is shifted to obtain each the following graphs. What is the equation of the function, g(x), for each graph? Write your answers in terms of x3 or x.

a)

b)

8) Consider the following graph of y = f(x).

a) If h(x) = f(x) + 3, what would the new coordinates of P be after the shift? Give answer in (x, y) form.

b) If , what would the new coordinates of P be after the reflection? Give answer in (x, y) form.

9) Consider the function .

a) Find h, the x-coordinate of the vertex of this parabola.

b) Substitute the two integers immediately to the left of h and the two integers immediately to the right of h into the function to find the corresponding y. Fill in the following table. Make sure your x-values are in increasing order in your table.

x y

h =__

c) Use MS Excel or another web-based graphing utility to graph the function by plotting the points found in the table in part b. Paste your graph here.

10) Find the equations for the horizontal and vertical asymptotes of the following. Type none if the function does not have an asymptote.

a)
Horizontal:
Vertical:

b)
Horizontal:
Vertical:

c)

Horizontal:
Vertical:

d)
Horizontal:
Vertical:

[See the attached questions file.]

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