This exam was adminstered in January 2025.
More Regents problems.
January 2025 Algebra Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
1. When factored, the expression x3 - 36x is equivalent to
(1) (x + 6)(x - 6)
(2) (x + 18)(x - 18)
(3) x(x + 6)(x - 6)
(4) x(x + 18)(x - 18)
Answer: (3) x(x + 6)(x - 6)
If the expression is third-degree, that is, it has x3 in it, then there must be three factors of x when factored. Eliminated choices (1) and (2).
The Greatest Common Factor of the two terms is x. When you factor the x, you get x(x2 - 36).
That binomial is the difference of two perfect squares which factors into (x + 6)(x - 6) because (6)(-6) equals -36 and (6) + (-6) = 0.
The correct answer is Choice (3).
You can put y = x3 - 36x into your graphing calculator and then graph the four choices. You will see that the Choices (1) and (2) are parabolas and that Choice (4) does not match the original graph. You can also look at the Table of Values if the graph goes off the screen.
2. Which equation represents the line that passes through the points (-1,8) and (4,-2)?
(1) y = -2x + 6
(2) y = -2x + 10
(3) y = -0.5x + 7.5
(4) y = -0.5x + 8.5
Answer: (1) y = -2x + 6
Find the slope of the line and then check which choice has an equation that works for both points.
The slope of a line is (y2 - y1) / (x2 - x1) = (-2 - 8) / (4 - (-1)) = 10 / (-5) = -2. Eliminate Choices (3) and (4).
At this point, you can graph the first two lines and check the Table of Values. Or you can do it algebraically.
y - (-2) = -(2)(x - 4)
y - (-2) = -2x + 8
y = -2x + 6
Once again, you could have graph all four lines and checked the Table of Values to find a match.
The correct answer is Choice (1).
What is the common ratio?
3. A geometric sequence is shown below.
(1) 1/4
(2) 2
(3) 1/2
(4) 4
Answer: (4) 4
The common ratio is what you multiply each term by to get the next term. There are four 1/2's in 2, and 2 goes into 8 four times, etc.
The correct Choice is Choice (4).
If the common ratio was a fraction, like, 1/2 or 1/4, then the sequence would be getting smaller.
If the common ratio was 2, the sequence would've been 1/2, 1, 2, 4, 8, 16, 32, ...
4. What is the constant term of the polynomial 2x3 - x + 5 + 4x2?
(1) 5
(2) 2
(3) 3
(4) 4
Answer: (1) 5
The constant term is the one without a variable. It's a constant because its value doesn't change!
The constant in this polynomial is the + 5.
Choice (1) is the correct answer.
5. A landscaping company charges a set fee for a spring cleanup, plus
an hourly labor rate. The total cost is modeled by the function C(x) = 55x + 80. In this function, what does the 55 represent?
(1) the set fee for the cleanup
(2) the hourly labor rate for a cleanup
(3) the profit earned by the company for one cleanup
(4) the number of hours of labor required for one cleanup
Answer: (2) the hourly labor rate for a cleanup
The variable represents the number of hours worked. That is the only thing unknown. The variable is multiplied by the hourly labor rate, which in this function is 55.
The correct answer is (2).
6. Which expression is equivalent to (5x2 - 2x + 4) - (3x2 + 3x - 1)?
(1) 2x2 + x + 3
(2) 2x2 - 5x + 5
(3) 2x4 + x2 + 3
(4) 2x4 - 5x2 + 5
Answer: (1) 2x2 + x + 3
When adding or subtracting polynommials, the exponents DO NOT change. You can only combine like terms, like counting puppies and kittens. Eliminate Choices (3) and (4).
Distribute the minus sign and combine like terms.
(5x2 - 2x + 4) - (3x2 + 3x - 1)
5x2 - 2x + 4 - 3x2 - 3x + 1
2x2 - 5x + 3
The correct answer is Choice (1).
Which point is a solution to this system?
7. A system of inequalities is graphed on the set of axes below.
(1) (1,1)
(2) (2,-2)
(3) (1,8)
(4) (4,2)
Answer: (4) (4,2)
The solution to the system of inequalities is the set of points in the double-shaded (criss-cross) area as long as they are NOT on the broken line. The broken line is not in the solution set. It represents a boundary that isn't part of the solution.
Choice (1) is the intersection of the broken and solid lines. Eliminate Choice (1).
Choice (2) is on the broken line. Eliminate Choice (2).
Choice (3) is a solution to only one of the two inequalities. Eliminate Choice (3).
Choice (4) is in the double-shaded area. This is the correct answer.
8. In an arithmetic sequence, the first term is 25 and the third term is 15.
What is the tenth term in this sequence?
(1) -20
(2) -25
(3) 70
(4) 75
Answer: (1) -20
The sequence is getting smaller, not bigger, so 70 and 75 are just silly. Eliminate Choices (3) and (4).
The difference is -10 after 2 terms, so each term is going down by 5. You could easily write this out:
25, 20, 15, 10, 5, 0, -5, -10, -15, -20.
The correct choice is Choice (1).
Algebraically, if each term subtracts 5 and the tenth term is nine terms after the first term, then the tenth term must be:
This is useful if you need to find a bigger number or if the common difference isn't an "easy" number like 5.
More to come. Comments and questions welcome.
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