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
In a survey of 300 people who were randomly sampled from a well-defined population, 60 said that they read a newspaper daily. If 1,000 people had been randomly sampled from the same population and asked the same question, how many would be expected to say they read a newspaper daily?
- A. 180
- B. 200
- C. 360
- D. 600
- E. 760
Correct Answer & Rationale
Correct Answer: A
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
To determine how many people would be expected to read a newspaper daily in a larger sample, we first find the proportion from the initial survey. Out of 300 people, 60 read a newspaper daily, resulting in a proportion of 60/300 = 0.2 or 20%. Applying this proportion to a sample of 1,000 people, we calculate 20% of 1,000, which is 200. Therefore, option B (200) is the expected number. Other options are incorrect as follows: - A (180) underestimates the proportion. - C (360) overestimates, assuming a higher reading rate. - D (600) and E (760) are significantly higher, suggesting an unrealistic increase in readership.
What is the sum of the two polynomials? 4x² + 3x + 5 + x² + 6x - 3?
- A. 4x² + 9x + 2
- B. 5x² + 9x + 2
- C. 5x² + 9x + 8
- D. 4x² + 9x² + 2
- E. 5x² + 9x² + 8
Correct Answer & Rationale
Correct Answer: B
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.
To find the sum of the polynomials \(4x^2 + 3x + 5\) and \(x^2 + 6x - 3\), we combine like terms. 1. For \(x^2\) terms: \(4x^2 + x^2 = 5x^2\). 2. For \(x\) terms: \(3x + 6x = 9x\). 3. For constant terms: \(5 - 3 = 2\). Thus, the resulting polynomial is \(5x^2 + 9x + 2\), which corresponds to option B. Option A incorrectly adds the \(x^2\) terms, leading to an incorrect polynomial. Option C miscalculates the constant term. Option D mistakenly adds the \(x^2\) terms incorrectly and does not follow proper polynomial addition. Option E also miscalculates by incorrectly summing the \(x^2\) terms and the constants.
What are the solutions to (x-2)(x+4) = 0?
- A. -4 and 2
- B. -3 and 1
- C. -2 and 4
- D. -1 and 1
- E. -1 and 3
Correct Answer & Rationale
Correct Answer: A
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
What are the solutions to the equation: x² - 10?
- A. ±5
- B. ±√10
- C. ±10
- D. ±10²
- E. ±20
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.