January 2024 Algebra 2, Part II

This exam was adminstered in January 2024.

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Algebra 2 January 2024

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Factor the expression x³ + 4x² - 9x - 36 completely.

Answer:

Factor by grouping, and then factor the quadratic you get after the first step.

There are two ways to group, and either should work in any question of this kind.

x³ + 4x² - 9x - 36
(x³ + 4x²) - (9x + 36)
x²(x + 4) - 9(x + 4)
(x² - 9)(x + 4)
(x + 3)(x - 3)(x + 4)

You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.

x³ - 9x + 4x² - 36
(x³ - 9x) + (4x² - 36)
x(x² - 9) + 4(x² - 9)
(x + 4)(x² - 9)
(x + 4)(x + 3)(x - 3)

Note: This was Very Similar to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3).

26. Determine if x + 4 is a factor of 2x³ + 10x² + 4x - 16. Explain your answer.

Answer:

If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4.

2(-4)³ + 10(-4)² + 4(-4) - 16 = 0

Since the expression is equal to zero when x = -4, then (x + 4) must be a factor.

You could also solve this using polynomial division.

(x + 4) divides evenly, with no remainder, so it is a factor.

27. An initial investment of $1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)^4t, where t is the time in years.
Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7.

Answer:

Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer).

V(7) = 1000(1.01)⁴⁽⁷⁾ = 1321.29

V(2) = 1000(1.01)⁴⁽²⁾ = 1082.86

Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is $48 to the nearest dollar.

28. When ( 1 / ∛(y²) ) y⁴ is written in the form yⁿ, what is the value of n? Justify your answer.

Answer:

Use the laws of exponents to change the radical into a fraction. The combine the terms.

( 1 / ∛(y²) ) y⁴
( 1 / (y^2/3) y⁴
(y^-2/3) y⁴
y^10/3

n = 10/3.

29. The heights of the members of a ski club are normally distributed. The average height is 64.7 inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall.

Answer:

They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this.

You need to use the normalcdf function.

Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%.

All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation.

30. The explicit formula a_n = 6 + 6n represents the number of seats in each row in a movie theater, where n represents the row number. Rewrite this formula in recursive form.

Answer:

A recursive function needs an initial value (a₁) and an equation for a_n is terms of a_n-1.

The inition value a₁ = 12.

Then a_n = a_n-1 + 6, because the common difference (rate of change) is 6.

31.Write (2xi³ - 3y)²) in simplest form.

Answer:

Square the binomial, substitute the powers of i, and Combine Like Terms.

(2xi³ - 3y)²)

(2xi³ - 3y)(2xi³ - 3y)

4x²i⁶ - 6xyi³ - 6xyi³ + 9y²

-4x² - 12xyi³ + 9y²

-4x² + 12xyi + 9y²

32. A survey was given to 1250 randomly selected high school students at the end of their junior year. The survey offered four post-graduation options: two-year college, four-year college, military, or work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can be made about the population of high school juniors, based on this survey

Answer:

This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%.

One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option.

End of Part II

How did you do?

More to come. Comments and questions welcome.

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