Algebra Problems of the Day (Algebra Regents, August 2024 Part I)

This exam was adminstered in August 2024.

More Regents problems.

August 2024 Algebra Regents

Part I

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


17. If x = 4a² - a + 3 and y = a - 5, then which polynomial is equivalent to the product of x and y?

(1) -17a² - 2a - 15
(2) -17a² + 8a - 15
(3) 4a³ - 21a² - 2a - 15
(4) 4a³ - 21a² + 8a - 15

Answer: (3) 4a³ - 21a² - 2a - 15

First, if 4a² is multiplied by a, then the leading term of the product must be 4a³. Eliminate Choices (1) and (2).

(4a² - a + 3) (a - 5)
4a³ - 20a² - a² - 5a + 3a - 15
4a³ - 21a² - 2a - 15

This is Choice (3), which is the correct answer.

18. What is an equation of the line that passes through (3,7) and has a slope of 2?

(1) y - 7 = 2(x - 3)
(2) y - 3 = 2(x - 7)
(3) y + 7 = 2(x + 3)
(4) y + 3 = 2(x + 7)

Answer: (1) y - 7 = 2(x - 3)

Point-slope form of a variable is y - y₀ = m(x - x₀), with minus signs in front of the (x₀, y₀).

This is Choice (1).

Choices (2) and (4) mix up the x- and y-coordinates.

Choice (3) has plus signs instead of minus signs.

You can graph it by adding 7 to both sides of the equation: y = 2(x - 3) + 7, which will show a graph with a slope of 2, going through the point (3,7).

19. A geometric sequence with a common ratio of -3 is

(1) -10, -7, -4, -1, ...
(2) 14, 11, 8, 5, ...
(3) -2, -6, -18, -54, ...
(4) 4, -12, 36, -108, ...

Answer: (4) 4, -12, 36, -108, ...

A geometric sequence with a common ratio of -3 would be switching signs from positive to negative to positive with each successive term. (A positive number times -3 becomes negative, and a negative number times -3 becomes positive.) The only possibility is Choice (4).

In Choice (1), there is no common ratio. There is a common difference of +3. (-7 - -10 = 3, -4 - -7 = 3, ...)

In Choice (2), there is no common ratio. There is a common difference of -3. (11 - 14 = 3, 8 - 11 = 3, ...)

In Choice (3), there is a common ratio of +3. Each number is three times the sizes of the previous term. (-6/-2 = 3, -18/-6 = 3, ...)

In Choice (4), there is a common ratio of -3. This is the correct choice. (-12/4 = -3, 36/-12 = -3, ...)

20. When the equation 6 - ax = ax - 2 is solved for x in terms of a, and a =/= 0, the result is

(1) 4a
(2) 4/a
(3) 2a
(4) 2/a

Answer: (2) 4/a

Use inverse operations to isolate the variable x.

6 - ax = ax - 2
8 = 2ax
8/(2a) = x
4/a = x

Choice (2) is the correct answer.

The clause that "a =/= 0" was a bit of a hint that the correct choice would probably be one of the two with "/a" in them.

21. Which function has the zeros -1, 3, and -4?

(1) f(x) = (x + 1)(x - 3)(x - 4)
(2) g(x) = (x - 1)(x + 3)(x - 4)
(3) h(x) = (x + 1)(x - 3)(x + 4)
(4) k(x) = (x - 1)(x + 3)(x + 4)

Answer: (3) h(x) = (x + 1)(x - 3)(x + 4)

The zeroes of the function are the values of x which make one of the factors equal to zero.

If x = -1, then -1 + 1 = 0. If x = 3, then 3 - 3 = 0. If x = -4 then x + 4 = 0.

Choice (3) is the correct answer.

If you are confused, you can graph each of these functions in your graphing calculator and check which one has the correct zeroes.

22. The expression 5^{a + 2b} is equivalent to

(1) 5^a * 5² * 5^b
(2) 5^a * 25^b
(3) 25^2ab
(4) 25^a+2b

Answer: (2) 5^a * 25^b

The rules for exponents are that you add the exponents when you multiply a base with one exponent times the same base with an exponent (same or different), but you multiply the exponents when you have a base with an exponent raised to another exponent.

The expression

5^{a + 2b} is equivalent to 5^a * 5^2b and 5^a * (5²)^b

Choice (1) is equivalent to 5^{a + 2 + b}. Eliminate Choice (1).

Choice (2) is equivalent to 5^a * (5²)^b. This is the correct answer.

Choice (3) is equivalent to (5²)^2ab. Eliminate Choice (3).

Choice (4) is equivalent to (5²)^{a + 2b}. Eliminate Choice (4).

23. In an arithmetic sequence, the first term is 4 and the third term is -2. What is the common difference?

(1) -1
(2) -2
(3) -3
(4) -6

Answer: (3) -3

The common difference is (-2 - 4) / (3 - 1), which is similar to the slope formula.

The common difference is -3, which is Choice (3).

If you look at each of the choices:

4 - 1 = 3, 3 - 1 = 2. Eliminate Choice (1).

4 - 2 = 2, 2 - 2 = 0. Eliminate Choice (2).

4 - 3 = 1, 1 - 3 = -2. This is the correct answer.

4 - 6 = -2, -2 - 6 = -8. Eliminate Choice (4).

24. Joe is ordering water for his swimming pool. He determines the volume of his pool to be about 3240 cubic feet. There are approximately 7.5 gallons of water in 1 cubic foot. A truck load holds 6000 gallons of water. Which expression would allow Joe to correctly calculate the number of truck loads of water he needs to fill his pool?

Answer: (4) See image

The final expression has to have "truck loads" per pool as its unit, and all the other units need to cancel out. That means that gallons has to appear once in the numerator and once in the denomimator, that ft³ must appear once in the numerator and once in the denominator, and "truck load" must appear only in the numerator. "Pool" will be in the denominator.

The first choice has ft⁶ per (pool * truck load). Eliminate Choice (1).

The second choice has ft⁶ * truck load per (pool * gal ²). Eliminate Choice (2).

The third choice has gal² per (pool * truck load). Eliminate Choice (3).

The fourth choice has truck load per pool. This is the correct answer.

End of Part I.
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