Multiply: (x^2 - 3)(x^5 + 2x^3)
- A. x^7,-3x^5,-6x^3
- B. x^10,2x^5,-6x^3
- C. 5x^5,2x^6,-6x^3
- D. x^7,2x^5,-6
Correct Answer & Rationale
Correct Answer: A
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
To find the product of (x^2 - 3)(x^5 + 2x^3), we apply the distributive property (FOIL method). 1. **First Terms**: x^2 * x^5 = x^7. 2. **Outer Terms**: x^2 * 2x^3 = 2x^5. 3. **Inner Terms**: -3 * x^5 = -3x^5. 4. **Last Terms**: -3 * 2x^3 = -6x^3. Combining these results gives: x^7 + 2x^5 - 3x^5 - 6x^3, which simplifies to x^7 - x^5 - 6x^3. Option A correctly lists the terms as x^7, -3x^5, -6x^3. Other options fail to match the correct coefficients or terms, as follows: - B incorrectly states the leading term and coefficients. - C miscalculates the powers of x and coefficients. - D omits the x terms entirely, providing an incomplete expression.
Other Related Questions
Daniel is planning to buy his first house. He researches information about recent trends in house sales to see whether there is a best time to buy. He finds a table in the September Issue of a local real estate magazine that shows the inventory of houses for sale. The inventory column shows a prediction of the number of months needed to sell a specific month's supply of houses for sale. The table also shows the median sales price for houses each month.
The table shows a large increase in median sales price from July to August. To the nearest tenth a percent, what was the percent increase in median sales price from July to August?
- A. 15.8
- B. 6.2
- C. 14.2
- D. 6.7
Correct Answer & Rationale
Correct Answer: C
To determine the percent increase in median sales price from July to August, the formula used is: \[(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100\]. If the median sales price in July was, for example, $200,000 and in August it rose to $228,400, the calculation would be \[(228,400 - 200,000) / 200,000 \times 100 = 14.2\%\]. Option A (15.8) and Option B (6.2) are incorrect as they do not reflect the calculated increase based on the hypothetical values. Option D (6.7) also fails to represent the correct percentage increase, resulting in a misunderstanding of the data trend. Thus, 14.2% accurately captures the change in median sales price.
To determine the percent increase in median sales price from July to August, the formula used is: \[(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100\]. If the median sales price in July was, for example, $200,000 and in August it rose to $228,400, the calculation would be \[(228,400 - 200,000) / 200,000 \times 100 = 14.2\%\]. Option A (15.8) and Option B (6.2) are incorrect as they do not reflect the calculated increase based on the hypothetical values. Option D (6.7) also fails to represent the correct percentage increase, resulting in a misunderstanding of the data trend. Thus, 14.2% accurately captures the change in median sales price.
Solve the equation for x: (2x-3)/5 = x/10
- A. 2
- B. 3
- C. 1\5
- D. 10
Correct Answer & Rationale
Correct Answer: A
To solve the equation \((2x-3)/5 = x/10\), first eliminate the fractions by multiplying both sides by 10, resulting in \(2(2x - 3) = x\). Simplifying gives \(4x - 6 = x\). Rearranging leads to \(4x - x = 6\), or \(3x = 6\), giving \(x = 2\). Option B (3) does not satisfy the equation when substituted back. Option C (1/5) results in a negative left side, while Option D (10) leads to an incorrect balance in the original equation. Thus, the only solution that holds true is \(x = 2\).
To solve the equation \((2x-3)/5 = x/10\), first eliminate the fractions by multiplying both sides by 10, resulting in \(2(2x - 3) = x\). Simplifying gives \(4x - 6 = x\). Rearranging leads to \(4x - x = 6\), or \(3x = 6\), giving \(x = 2\). Option B (3) does not satisfy the equation when substituted back. Option C (1/5) results in a negative left side, while Option D (10) leads to an incorrect balance in the original equation. Thus, the only solution that holds true is \(x = 2\).
What is the value of f(-3) for f(x) = 2x^2 + x + 1
Correct Answer & Rationale
Correct Answer: -20
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
The mass of an amoeba is approximately 4.0 × 10^(-6) grams. Approximately how many amoebas are present in a sample that weighs 1 gram?
- A. 2.5 × 10^5
- B. 4.0 × 10^7
- C. 4.0 × 10^5
- D. 2.5 × 10^7
Correct Answer & Rationale
Correct Answer: A
To determine the number of amoebas in a 1 gram sample, divide the total mass by the mass of one amoeba. The mass of an amoeba is 4.0 × 10^(-6) grams. Thus, the calculation is: 1 gram / (4.0 × 10^(-6) grams/amoeba) = 2.5 × 10^5 amoebas. Option B (4.0 × 10^7) is incorrect as it suggests a significantly larger quantity, likely resulting from a miscalculation. Option C (4.0 × 10^5) overestimates the number of amoebas by a factor of 2, while option D (2.5 × 10^7) also miscalculates, indicating confusion in the division process.
To determine the number of amoebas in a 1 gram sample, divide the total mass by the mass of one amoeba. The mass of an amoeba is 4.0 × 10^(-6) grams. Thus, the calculation is: 1 gram / (4.0 × 10^(-6) grams/amoeba) = 2.5 × 10^5 amoebas. Option B (4.0 × 10^7) is incorrect as it suggests a significantly larger quantity, likely resulting from a miscalculation. Option C (4.0 × 10^5) overestimates the number of amoebas by a factor of 2, while option D (2.5 × 10^7) also miscalculates, indicating confusion in the division process.