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ALGEBRA I REGENTS HIGH SCHOOL EXAMINATION 2025

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NYS Regents Algebra I Exam - January 2025

Official examination document from nysedregents.org

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Question 1: Factoring Problem Solution

Question 1:

When factored, the expression x3 - 36x is equivalent to?

Step-by-Step Solution

Step 1: Find the Greatest Common Factor (GCF)

The first step in factoring any expression is to look for a common factor in all its terms. In the expression x3 - 36x, both terms share a factor of x.

We can factor out this GCF:

x(x2 - 36)

Step 2: Factor the Difference of Squares

Now, examine the expression inside the parentheses: x2 - 36. This is a special pattern known as a "difference of squares," which follows the formula a2 - b2 = (a + b)(a - b).

  • In our expression, a2 is x2, so a = x.
  • And b2 is 36, so b = 6.

Applying the difference of squares formula, we get:

(x + 6)(x - 6)

Step 3: Combine the Factors

To get the final, fully factored expression, we combine the GCF from Step 1 with the factored binomials from Step 2.

x(x + 6)(x - 6)

Conclusion

The completely factored expression x(x + 6)(x - 6) matches option (3) from the choices provided.

The correct answer is (3) x(x + 6)(x - 6)

Question 2: Finding the Equation of a Line

Question 2:

Which equation represents the line that passes through the points (-1, 8) and (4, -2)?

Step-by-Step Solution

Step 1: Calculate the Slope (m)

To find the equation of a line, we first need its slope. The formula for the slope (m) between two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ - y₁) / (x₂ - x₁)

Let's plug in our points (-1, 8) and (4, -2):

m = (-2 - 8) / (4 - (-1)) = -10 / 5 = -2

The slope of the line is -2. This eliminates options (3) and (4), which have a slope of -0.5.

Step 2: Find the Y-intercept (b)

Now we use the slope-intercept form of a line, y = mx + b, where `b` is the y-intercept. We can plug in our slope (m = -2) and one of the given points to solve for `b`. Let's use the point (-1, 8).

Substitute y=8, m=-2, and x=-1 into the equation:

8 = (-2)(-1) + b

8 = 2 + b

8 - 2 = b

6 = b

The y-intercept (b) is 6.

Step 3: Write the Final Equation

With the slope (m = -2) and the y-intercept (b = 6), we can write the full equation of the line:

y = -2x + 6

Conclusion

The calculated equation y = -2x + 6 matches option (1) from the choices provided.

The correct answer is (1) y = -2x + 6

Question 3: Finding the Common Ratio

Question 3:

A geometric sequence is shown below. What is the common ratio?

1/2, 2, 8, 32, ...

Step-by-Step Solution

Step 1: Understand the Common Ratio

In a geometric sequence, each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio (r).

To find the common ratio, you can divide any term by its preceding term. The formula is:

r = (a term) / (the previous term)

Step 2: Calculate the Ratio

Let's take the second term (2) and divide it by the first term (1/2):

r = 2 / (1/2) = 2 × 2 = 4

Step 3: Verify the Ratio

To confirm that the ratio is "common," we can check it with another pair of consecutive terms. Let's divide the third term (8) by the second term (2):

r = 8 / 2 = 4

Since the result is the same, we can be confident that the common ratio is 4.

Conclusion

The common ratio for the sequence is 4. This matches option (4) from the choices provided.

The correct answer is (4) 4

Question 4: Identifying the Constant Term

Question 4:

What is the constant term of the polynomial 2x³ - x + 5 + 4x²?

Step-by-Step Solution

Step 1: Understand What a Constant Term Is

In a polynomial, the constant term is the term that does not contain any variables (like `x`). Its value does not change, which is why it's called "constant."

Step 2: Examine the Terms of the Polynomial

Let's look at each term in the given polynomial: 2x³ - x + 5 + 4x².

  • 2x³: This term has a variable (`x³`).
  • -x: This term has a variable (`x`).
  • +5: This term has no variable.
  • +4x²: This term has a variable (`x²`).

Step 3: Identify the Constant

By looking at the list above, the only term without a variable is 5. Therefore, 5 is the constant term of the polynomial.

Note: The order of the terms doesn't matter. Whether the polynomial is written as 2x³ - x + 5 + 4x² or in standard form as 2x³ + 4x² - x + 5, the constant term remains 5.

Conclusion

The constant term of the polynomial is 5. This matches option (1) from the choices provided.

The correct answer is (1) 5

Question 5: Interpreting a Linear Function

Question 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?

Step-by-Step Solution

Step 1: Analyze the Structure of the Function

The function C(x) = 55x + 80 is a linear equation, which follows the general form y = mx + b.

  • y (or C(x)) is the total cost.
  • m is the slope, or the rate of change.
  • x is the variable (in this case, the number of hours).
  • b is the y-intercept, or the initial, fixed amount.

Step 2: Relate the Function to the Problem Description

The problem states there are two parts to the cost:

  • A "set fee": This is a fixed, one-time charge that does not change. In the equation, this corresponds to the constant term, or y-intercept, which is 80.
  • An "hourly labor rate": This is a variable cost that depends on the number of hours worked. In the equation, this is represented by the term 55x.

Step 3: Interpret the Number 55

In the term 55x, the number 55 is multiplied by x, which represents the number of hours. This means that for every hour of labor, the total cost increases by 55. Therefore, 55 represents the cost per hour.

This directly corresponds to the "hourly labor rate".

Conclusion

The number 55 is the rate of change in the cost function, representing the amount charged for each hour of labor. This matches option (2) from the choices provided.

The correct answer is (2) the hourly labor rate for a cleanup

Question 6: Subtracting Polynomials

Question 6:

Which expression is equivalent to (5x² - 2x + 4) - (3x² + 3x - 1)?

Step-by-Step Solution

Step 1: Distribute the Negative Sign

The first step is to distribute the subtraction sign to every term in the second parenthesis. This changes the sign of each term inside (3x² + 3x - 1).

(5x² - 2x + 4) - (3x² + 3x - 1)

↓

5x² - 2x + 4 - 3x² - 3x + 1

Step 2: Group Like Terms

Now, rearrange the expression to group terms with the same variable and exponent together.

(5x² - 3x²) + (-2x - 3x) + (4 + 1)

Step 3: Combine Like Terms

Finally, perform the addition or subtraction for each group of like terms.

  • 5x² - 3x² = 2x²
  • -2x - 3x = -5x
  • 4 + 1 = 5

Combining these results gives us the final simplified expression:

2x² - 5x + 5

Conclusion

The simplified expression is 2x² - 5x + 5. This matches option (2) from the choices provided.

The correct answer is (2) 2x² - 5x + 5

Question 7: Solving a System of Inequalities Graphically

Question 7:

A system of inequalities is graphed on the set of axes below. Which point is a solution to this system?

Step-by-Step Solution

Step 1: Identify the Solution Region

A solution to a system of inequalities is any point that satisfies all inequalities in the system simultaneously. On a graph, the solution set is the region where the shaded areas of all individual inequalities overlap.

In the given graph, this is the cross-hatched region where both types of shading are present.

Step 2: Analyze the Boundary Lines

It's also important to check the boundary lines themselves:

  • A solid line means points on the line are included in the solution for that inequality.
  • A dashed line means points on the line are not included in the solution.

Therefore, any point that lies on the dashed line cannot be a solution to the system.

Step 3: Test Each Point

We need to check which of the given points falls into the cross-hatched solution region.

  • (1, 1): This point is located in the cross-hatched region where the shading overlaps. It is a solution.
  • (2, -2): This point lies directly on the dashed line. Since the line is dashed, this point is not a solution.
  • (1, 8): This point is in a region with only one type of shading, not in the overlapping section. It is not a solution.
  • (4, 2): This point is also in a region with only one type of shading, not in the overlapping section. It is not a solution.

Conclusion

Only the point (1, 1) is located in the region where the solutions to both inequalities overlap. This matches option (1).

The correct answer is (1) (1,1)

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