What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?
- A. x³ - 8x² + 21x - 30
- B. x³ - 8x² - 21x - 30
- C. x³ - 8x² - 9x - 30
- D. x³ + 8x² + 21x + 30
- E. x³ + 8x² - 9x + 30
Correct Answer & Rationale
Correct Answer: A
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
Other Related Questions
Each month, the charge for a lawn care service consists of a flat fee of $25, plus $5 each time the lawn is mowed. Which of the following equations represents the total monthly charge, A(m), in dollars, as a function of the number of times the lawn is mowed, m?
- A. A(m) = 5(25)m
- B. A(m) = 5 + 25m
- C. A(m) = 5m + 25
- D. A(m) = 25m + 5
- E. A(m) = m + 5 + 25
Correct Answer & Rationale
Correct Answer: C
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
The equation A(m) = 5m + 25 accurately represents the total monthly charge for the lawn care service. Here, the term 5m accounts for the $5 charge per mowing, and the flat fee of $25 is added to this total. Option A incorrectly multiplies the flat fee by the number of mowings, which misrepresents the structure of the charges. Option B misplaces the flat fee, summing it with the number of mowings instead of adding it as a fixed cost. Option D incorrectly places the flat fee as a coefficient of m, which distorts the relationship. Option E combines the charges incorrectly, failing to clearly separate the flat fee from the per-mow charge.
Square PQRS, with a side length of 5 units, will be translated 2 units to the right and 2 units up in the standard (x, y) coordinate plane. What is the area, in square units, of the image of PQRS?
- A. 20
- B. 25
- C. 40
- D. 50
- E. 100
Correct Answer & Rationale
Correct Answer: B
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
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.
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.