This exam was adminstered in January 2024.

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__Algebra 2 January 2024__

__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 x ^{3} + 4x^{2} - 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.

x^{3} + 4x^{2} - 9x - 36

(x^{3} + 4x^{2}) - (9x + 36)

x^{2}(x + 4) - 9(x + 4)

(x^{2} - 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.

x^{3} - 9x + 4x^{2} - 36

(x^{3} - 9x) + (4x^{2} - 36)

x(x^{2} - 9) + 4(x^{2} - 9)

(x + 4)(x^{2} - 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 2x ^{3} + 10x^{2} + 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 / ∛(y ^{2}) ) y^{4} is written in the form y^{n}, 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 / ∛(y^{2}) ) y^{4}

( 1 / (y^{2/3}) y^{4}

(y^{-2/3}) y^{4}

y^{10/3}

n = 10/3.

*29. The heights of the members of a ski club are normally distributed. The average height is 64.7inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to thenearest 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 (a_{1}) and an equation for a_{n} is terms of a_{n-1}.

The inition value a_{1} = 12.

Then a_{n} = a_{n-1} + 6, because the common difference (rate of change) is 6.

*31.Write (2xi ^{3} - 3y)^{2}) in simplest form.*

**Answer: **

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

(2xi^{3} - 3y)^{2})

(2xi^{3} - 3y)(2xi^{3} - 3y)

4x^{2}i^{6} - 6xyi^{3} - 6xyi^{3} + 9y^{2}

-4x^{2} - 12xyi^{3} + 9y^{2}

-4x^{2} + 12xyi + 9y^{2}

*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, orwork. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that canbe 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

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