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.
Other Related Questions
Josh takes 6 hours to paint a room. Margaret can paint the same room in 4 hours. Assuming their individual rates do not change, how long will it take them to paint the room together?
- A. 1.5 hours
- B. 2.4 hours
- C. 4.8 hours
- D. 5 hours
- E. 10 hours
Correct Answer & Rationale
Correct Answer: B
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
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.
The relationship between h, a person's height in inches, and f, the length in inches of the person's femur, is modeled by the equation: h = 1.88f + 32. Which statement correctly identifies and describes the slope of the equation?
- A. The slope of the equation is 1.88, and it represents the femur length, in inches, when the height is 32 inches.
- B. The slope of the equation is 1.88, and it represents the number of inches the height increases for each inch the femur length increases
- C. The slope of the equation is 1.88, and it represents the number of inches the femur length increases for each inch the height increases
- D. The slope of the equation is 32, and it represents the number of inches the height increases for each inch the femur length increases.
- E. The slope of the equation is 32, and it represents the height, in inches, when the femur length is 1.88 inches.
Correct Answer & Rationale
Correct Answer: B
The slope of 1.88 in the equation h = 1.88f + 32 indicates that for every additional inch in femur length (f), height (h) increases by 1.88 inches. This relationship highlights the direct impact of femur length on height. Option A misinterprets the slope, incorrectly stating it represents femur length at a specific height. Option C reverses the relationship, suggesting femur length increases with height, which is inaccurate. Option D incorrectly identifies the slope as 32 and misrepresents the relationship. Option E also incorrectly identifies the slope and misinterprets its meaning in the context of the equation.
The slope of 1.88 in the equation h = 1.88f + 32 indicates that for every additional inch in femur length (f), height (h) increases by 1.88 inches. This relationship highlights the direct impact of femur length on height. Option A misinterprets the slope, incorrectly stating it represents femur length at a specific height. Option C reverses the relationship, suggesting femur length increases with height, which is inaccurate. Option D incorrectly identifies the slope as 32 and misrepresents the relationship. Option E also incorrectly identifies the slope and misinterprets its meaning in the context of the equation.
An irrigation pivot makes a circle with a radius of about 400 meters. Which of the following values is closest to the area, in square meters, of the circle?
- A. 1300
- B. 2500
- C. 160000
- D. 502700
- E. 1579100
Correct Answer & Rationale
Correct Answer: D
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. With a radius of 400 meters, the area calculates to approximately \( A = \pi \times (400)^2 \approx 502700 \) square meters, making option D the closest value. Option A (1300) is far too low, indicating a misunderstanding of the formula. Option B (2500) is also significantly underestimated for such a large radius. Option C (160000) is closer but still incorrect, as it neglects the multiplication by \( \pi \). Option E (1579100) overestimates the area, suggesting a miscalculation of the radius or the area formula.