(x, why?): Algebra 2 Problems of the Day (Algebra 2, January 2025 Part I)

This exam was adminstered in January 2025.

January 2025 Algebra 2 Regents

Part I

Each correct answer will receive 2 credits. No partial credit.

1. The exact value of sin(8π/3) is

(1) 1/2
(2) -1/2
(3) -√(3)/2
(4) √(3)/2

Answer: (4) √(3)/2

Consider the unit circle, which has a length of 2π. The fraction 8/3 is equal to 2 and 2/3, so 8π/3 is the same as 2π/3, which is Quadrant II where sin is negative. Eliminate Choices (1) and (4).

The point on the unit circle is (-1/2, √(3)/2), where cos is the x value and sin is the y value.

The correct answer is Choice (4).

2. A teacher randomly divides all of her students into two groups. She grades the homework for one group but does not grade the homework for the other group. All homework is returned to the students. She then compares test scores for each of the groups to see if grading homework has an effect on the test scores.
This method of data collection is best described as

(1) an experiment
(2) an unbiased survey
(3) a simulation
(4) an observational study

Answer: (1) an experiment

This is an experiment. It isn't a survey or a simulation and it isn't observational in nature.

The correct answer is Choice (1).

3. Which expression is equivalent to (x - 2)² + 27(x - 2)² - 90?

(1) (x + 30)(x - 3)
(2) (x + 28)(x - 5)
(3) (x - 30)(x + 3)
(4) (x - 2)(x + 25)(x - 90)

Answer: (2) (x + 28)(x - 5) down

Replace (x - 2) with another variable, such as y, and factor the polynomial:

y² + 27y² - 90
(y + 30)(y - 3)
(x - 2 + 30)(x - 2 - 3)
(x + 28)(x - 5)

The correct answer is Choice (2).

4. Given the functions f(x) = 2x + 5/2 and g(x) = 3/x, what are the solutions to f(x) = g(x)?

(1) (0.75, 4) or (-2,-1.5)
(2) x = 0.75 or x = -2
(3) y = -1.5 or y = 4
(4) (-2, 0.75)

Answer: (2) x = 0.75 or x = -2

Note that all four choices have the same values, so you don't have to solve anything.

Finding solutions to f(x) = g(x) means "for what values of x are the two functions equal?" That means that you are looking for the x values, not the function values.

Choice (2) has the x-values and only the x-values, and this is the correct answer.

5. Given f(x) = 2x³ - 3x² - 5x - 12 and g(x) = x - 3, the quotient of f(x)/g(x) is

(1) 2x² + 3x + 4
(2) 2x³ + 3x² + 4x
(3) 2x² - 9x + 22 - 78/(x - 3)
(4) 2x³ - 9x² + 22x - 78

Answer: (1) 2x² + 3x + 4

If 2x³ is divided by (x - 3), the highest power in the quotient will be 2, so eliminate Choices (2) and (4).

Use polynomial long division to solve this:

This is Choice (1).

6. Abby is told that each day there is a 50% chance it will rain. Which simulation can Abby perform to determine the likelihood of it raining for the next seven days?

(1) Flip a coin seven times, count how many heads, and repeat 50 times.
(2) Roll a die seven times, count how many twos, and repeat 50 times.
(3) Roll a pair of dice, count totals of seven, and repeat 50 times
(4) Flip a coin 50 times and count how many heads.

Answer: (1) Flip a coin seven times, count how many heads, and repeat 50 times.

Running a simulation means running the test multiple times and analyzing the results. Choice (4) doesn't do that.

In Choice (1), there is a 50% chance that the coin will land heads, and the simulation is repeated 50 times. This is the correct answer.

In Choice (2), there is only a 1/6 probability of rolling a two, not 50%. Eliminate Choice (2).

In Choice (3), there is only a 1/6 probability of rolling a seven on two dice. Eliminate Choice (3).

7. What are the solutions to 4x² - 7x - 2 = -10?

(1) -1/4, 2
(2) 7/8 + √(79)/8 i
(3) 7/8 + √(241)/8
(4) 7/8 + √(143)/8 i

Answer: (2) 7/8 + √(79)/8 i

Set the equation equal to 0 and use the Quadratic Formula.

4x² - 7x + 8 = 0

x = ( -(-7) + √((-7)² - 4(4)(8) ) / (2(4))

x = ( 7 + √( 49 - 128 ) / (8))

x = ( 7 + √( -79 ) / (8))

x = 7/8 + i √(79)/8

This is Choice (2).

8. If x - 5 is a factor of p(x) = ax⁴ + bx³ + cx² + dx + e, then which statement must be true?

(1) p(-5) = 0
(2) p(-5) =/= 0
(3) p(5) = 0
(4) p(5) =/= 0

Answer: (3) p(5) = 0

If x - 5 is a factor of p(x), then by definition, p(5) = 0. This is Choice (3).

We don't know anything about p(-5).

More to come. Comments and questions welcome.

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