## Monday, February 19, 2024

### 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.

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.

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.

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.

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

( 1 / ∛(y2) ) y4
( 1 / (y2/3) y4
(y-2/3) y4
y10/3

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.

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 an = 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.

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

The inition value a1 = 12.

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

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

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

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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