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
A bowl contains 18 pieces of candy: 8 red, 6 orange, and 4 green. Brandon will select 1 piece of candy at random. What is the probability that Brandon will select a green piece?
- A. 2/7
- B. 2/9
- C. 2/11
- D. 1/9
- E. 1/8
Correct Answer & Rationale
Correct Answer: B
To find the probability of selecting a green piece of candy, divide the number of green candies by the total number of candies. There are 4 green candies and 18 total candies, resulting in a probability of 4/18, which simplifies to 2/9. Option A (2/7) incorrectly assumes a different total or count of green candies. Option C (2/11) suggests an inaccurate total of candies or green pieces. Option D (1/9) miscalculates the ratio of green candies to the total. Option E (1/8) also misrepresents the count of green candies. Only B accurately reflects the correct ratio.
To find the probability of selecting a green piece of candy, divide the number of green candies by the total number of candies. There are 4 green candies and 18 total candies, resulting in a probability of 4/18, which simplifies to 2/9. Option A (2/7) incorrectly assumes a different total or count of green candies. Option C (2/11) suggests an inaccurate total of candies or green pieces. Option D (1/9) miscalculates the ratio of green candies to the total. Option E (1/8) also misrepresents the count of green candies. Only B accurately reflects the correct ratio.
Which of the following equations does not represent y as a function of x in the standard (x, y) coordinate plane?
- A. y = x
- B. y = x + 2
- C. y = x² + 2
- D. x = y + 2
- E. x = y² + 2
Correct Answer & Rationale
Correct Answer: E
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
Option E, \( x = y^2 + 2 \), does not represent \( y \) as a function of \( x \) because it can yield multiple \( y \) values for a single \( x \) value. For example, when \( x = 6 \), \( y \) can be either 2 or -2, violating the definition of a function. In contrast, options A, B, and C express \( y \) explicitly in terms of \( x \), allowing only one output for each input. Option D, while rearranging the equation, can also be transformed into a function of \( y \) in terms of \( x \) (i.e., \( y = x - 2 \)). Thus, options A, B, C, and D all represent \( y \) as a function of \( x \).
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.
Which of the following statements is true about the graphs of f(x) = x and g(x) = 3x in the standard (x, y) coordinate plane?
- A. The graphs will not intersect.
- B. The graphs will intersect only at the point (0,0).
- C. The graphs will intersect only at the point (0,1).
- D. The graphs will intersect only at the point (1,1).
- E. The graphs will intersect only at the point (3,3).
Correct Answer & Rationale
Correct Answer: D
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).