This exam was adminstered in January 2024. More Regents problems. 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: There are two ways to group, and either should work in any question of this kind.
Algebra 2 January 2024
Factor by grouping, and then factor the quadratic you get after the first step.
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 / (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 64.7inches 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 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.
Answer:
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.
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, 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
How did you do?
More to come. Comments and questions welcome.
More Regents problems.
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