Which of the following expressions is equivalent to: 6x³ + 7x² + 1/x?
- A. 63 + 72 + 1/x
- B. 63 + 72 + 1
- C. 6x² + 7x + 1/x
- D. 6x² + 7x + 1
- E. 6x² + 7x² + 1
Correct Answer & Rationale
Correct Answer: C
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
Other Related Questions
The recommended dosage of a medicine is 4 milligrams per kilogram of body weight. What is the recommended dosage, in milligrams, for a person who weighs 84 kilograms?
- A. 21
- B. 88
- C. 324
- D. 336
- E. 2100
Correct Answer & Rationale
Correct Answer: D
To determine the recommended dosage for a person weighing 84 kilograms, multiply their weight by the dosage per kilogram: 4 mg/kg × 84 kg = 336 mg. Option A (21 mg) is incorrect as it significantly underestimates the dosage based on the weight. Option B (88 mg) also miscalculates the dosage, failing to apply the correct multiplication. Option C (324 mg) is close but still incorrect, as it does not reflect the accurate calculation. Option E (2100 mg) is far too high, indicating a misunderstanding of the dosage per kilogram. Thus, 336 mg is the correct dosage for the individual.
To determine the recommended dosage for a person weighing 84 kilograms, multiply their weight by the dosage per kilogram: 4 mg/kg × 84 kg = 336 mg. Option A (21 mg) is incorrect as it significantly underestimates the dosage based on the weight. Option B (88 mg) also miscalculates the dosage, failing to apply the correct multiplication. Option C (324 mg) is close but still incorrect, as it does not reflect the accurate calculation. Option E (2100 mg) is far too high, indicating a misunderstanding of the dosage per kilogram. Thus, 336 mg is the correct dosage for the individual.
The following is a list of triangles: I. Right triangles, II. Isosceles triangles, III. Equilateral triangles. A pair of triangles from which of these groups must be similar to each other?
- A. I only
- B. II only
- C. III only
- D. I and III only
Correct Answer & Rationale
Correct Answer: C
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Triangles from group III, equilateral triangles, are always similar to each other because they all have equal angles of 60 degrees, regardless of their size. Group I, right triangles, can vary significantly in angle measures beyond the right angle, so not all right triangles are similar. Similarly, group II, isosceles triangles, can have different base angles, leading to non-similar triangles. Thus, while right and isosceles triangles can share properties, only equilateral triangles guarantee similarity across the group. Therefore, option C accurately identifies the group with universally similar triangles.
Emma measured the height of her laptop screen. She reported the height as 8 inches, accurate to the nearest inch. The actual height of the screen must be:
- A. at least 7.5 inches and less than 8.5 inches
- B. at least 7.9 inches and less than 8.1 inches
- C. at least 7.99 inches and less than 8.01 inches
- D. at least 8 inches
- E. exactly 8 inches
Correct Answer & Rationale
Correct Answer: A
When measuring to the nearest inch, values can range from halfway to the next whole number. For Emma's reported height of 8 inches, this means the actual height must be at least 7.5 inches (inclusive) and less than 8.5 inches (exclusive). Option B is too narrow, only allowing for heights between 7.9 and 8.1 inches, which does not encompass all possible values. Option C is even more restrictive, only allowing for heights between 7.99 and 8.01 inches, excluding valid measurements. Option D is incorrect as it suggests the height must be 8 inches or more, which is too limiting. Option E incorrectly states the height must be exactly 8 inches, disregarding the range of possible values.
When measuring to the nearest inch, values can range from halfway to the next whole number. For Emma's reported height of 8 inches, this means the actual height must be at least 7.5 inches (inclusive) and less than 8.5 inches (exclusive). Option B is too narrow, only allowing for heights between 7.9 and 8.1 inches, which does not encompass all possible values. Option C is even more restrictive, only allowing for heights between 7.99 and 8.01 inches, excluding valid measurements. Option D is incorrect as it suggests the height must be 8 inches or more, which is too limiting. Option E incorrectly states the height must be exactly 8 inches, disregarding the range of possible values.
Let f(x) = 3x². What is f(-2x)?
- A. -36x²
- B. -12x²
- C. -6x²
- D. 12x²
- E. 36x²
Correct Answer & Rationale
Correct Answer: D
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.