answer:Step 1: Simplify and Understand the Equation Given that the quadratic polynomial ( f(x) ) satisfies the equation: [ (f(x))^3 - f(x) = 0, ] we can factorize it as follows: [ f(x) cdot (f(x) - 1) cdot (f(x) + 1) = 0. ] Step 2: Roots of the Factorized Equation The equation ( f(x) cdot (f(x) - 1) cdot (f(x) + 1) = 0 ) implies that: [ f(x) = 0 quad text{or} quad f(x) = 1 quad text{or} quad f(x) = -1. ] Step 3: Roots of the Quadratic Function Since ( f(x) ) is a quadratic polynomial, it can have at most 2 roots. Assuming ( f(x) = ax^2 + bx + c ), we know from the problem statement that the given equation has exactly three distinct solutions. Hence, we need to distribute these three roots among the three equations ( f(x) = 0 ), ( f(x) = 1 ), and ( f(x) = -1 ): 1. ( f(x) = 0 ) has at most 2 roots. 2. ( f(x) = 1 ) has at most 2 roots. 3. ( f(x) = -1 ) has at most 2 roots. Since no two of these equations share common roots and we need exactly 3 distinct solutions, one of these equations must have exactly one root. Step 4: Analyzing ( f(x) = 0 ) Assume ( f(x) = 0 ) has exactly one root. For a quadratic polynomial ( ax^2 + bx + c ) with exactly one root, it must be a perfect square: [ f(x) = a(x - h)^2, ] where ( h ) is the root of ( f(x) = 0 ). Step 5: Ordinate of the Vertex If ( f(x) = a(x - h)^2 ), the vertex form of the quadratic function shows that the ordinate (y-coordinate) of the vertex is: [ c = 0. ] This means the vertex is at the minimum (or maximum, depending on the sign of ( a )) value of the quadratic function, which is 0. Conclusion The ordinate of the vertex of the quadratic polynomial ( f(x) ) is: [ boxed{0}. ]

answer:Given the direction vector of line l is overrightarrow{d}=(1,sqrt{3}), we aim to find the inclination angle theta of line l. 1. The slope of the line, which is the tangent of the inclination angle theta, can be determined from the direction vector overrightarrow{d}=(1,sqrt{3}). This gives us the equation for the tangent of the angle: [ tan theta = frac{sqrt{3}}{1} = sqrt{3} ] 2. Knowing that tan theta = sqrt{3} and considering the range of theta in left[0,pi right) (since the inclination angle of a line is typically measured from the positive x-axis in a counter-clockwise direction), we can solve for theta. 3. From trigonometric identities, we know that tan frac{pi}{3} = sqrt{3}. Therefore, comparing this with our equation from step 1, we find: [ tan theta = tan frac{pi}{3} ] 4. This implies that theta = frac{pi}{3}, given the range for theta. Therefore, the inclination angle of line l is boxed{frac{pi}{3}}.

answer:To solve this problem, we need to calculate the probability that both Person A and Person B join clubs related to ball sports, given that they cannot join the same club and must choose from the 5 available clubs: "basketball," "soccer," "volleyball," "swimming," and "gymnastics." Out of these, only "basketball," "soccer," and "volleyball" are related to ball sports. **Step 1: Calculate the total number of ways Person A and Person B can join different clubs.** Since there are 5 clubs and each person must choose a different club, the total number of ways they can choose is the permutation of 5 items taken 2 at a time, which is calculated as: [ {A}_{5}^{2} = 5 times (5 - 1) = 20 ] **Step 2: Calculate the number of ways both can join clubs related to ball sports.** There are 3 clubs related to ball sports, so the number of ways both can choose from these clubs, without choosing the same club, is the permutation of 3 items taken 2 at a time: [ {A}_{3}^{2} = 3 times (3 - 1) = 6 ] **Step 3: Calculate the probability.** The probability that both join clubs related to ball sports is the ratio of the number of favorable outcomes to the total number of outcomes: [ text{Probability} = frac{{A}_{3}^{2}}{{A}_{5}^{2}} = frac{6}{20} = frac{3}{10} ] Therefore, the probability that both Person A and Person B join clubs related to ball sports, given that they join different clubs, is boxed{frac{3}{10}}.

answer:Since overrightarrow{a}=(3,5,1), overrightarrow{b}=(2,2,3), overrightarrow{c}=(4,-1,-3), The vector 2overrightarrow{a}-3overrightarrow{b}+4overrightarrow{c} =2(3,5,1)-3(2,2,3)+4(4,-1,-3) =(16,0,-19) Hence, the answer is boxed{(16,0,-19)}. Given the vectors overrightarrow{a}=(3,5,1), overrightarrow{b}=(2,2,3), overrightarrow{c}=(4,-1,-3), according to the scalar multiplication and vector addition formulas, we can find the coordinates of the vector 2overrightarrow{a}-3overrightarrow{b}+4overrightarrow{c}. This problem tests the knowledge of spatial vector coordinate operations. Mastering the scalar multiplication and vector addition formulas is the key to solving this problem.