answer:From Vieta's relations for the cubic equation, we know: [ a + b + c = 12, ] [ ab + bc + ca = 20, ] [ abc = 3. ] We need to find (frac{1}{a^2} + frac{1}{b^2} + frac{1}{c^2}). This can be expressed as: [ frac{(ab+bc+ca)^2 - 2abc(a+b+c)}{(abc)^2} ] By substituting the values from Vieta's relations: [ frac{(20)^2 - 2 cdot 3 cdot 12}{(3)^2} ] [ frac{400 - 72}{9} ] [ frac{328}{9} ] Thus, the result is (boxed{frac{328}{9}}).

answer:From the given information, we have f(x)=3ax^{2}+2bx+c. Since x_{1}, x_{2} are the two roots of the equation f(x)=0, we have x_{1}+x_{2}=- dfrac {2b}{3a}, x_{1}x_{2}= dfrac {c}{3a}. Thus, |x_{1}-x_{2}|^{2}=(x_{1}+x_{2})^{2}-4x_{1}⋅x_{2}= dfrac {4b^{2}-12ac}{9a^{2}}. Given that a+b+c=0, we have c=-a-b. Substituting this into the above equation, we get: |x_{1}-x_{2}|^{2}= dfrac {4b^{2}+12a(a+b)}{9a^{2}}= dfrac {12a^{2}+4b^{2}+12ab}{9a^{2}}= dfrac {4}{9}⋅( dfrac {b}{a})^{2}+ dfrac {4}{3}( dfrac {b}{a})+ dfrac {4}{3}①. Given that f(0)⋅f(1) > 0, we have (a+b)(2a+b) < 0, which implies 2a^{2}+3ab+b^{2} < 0. Since aneq 0, dividing both sides by a^{2}, we get: ( dfrac {b}{a})^{2}+3 dfrac {b}{a}+2 < 0, which leads to -2 < dfrac {b}{a} < -1. Substituting this into ①, we get |x_{1}-x_{2}|^{2}∈[ dfrac {1}{3}, dfrac {4}{9}). Hence, |x_{1}-x_{2}|∈[ dfrac { sqrt {3}}{3}, dfrac {2}{3}). Therefore, the answer is boxed{text{A}}. This problem tests the relationship between roots and coefficients, emphasizing the use of Vieta's theorem. The difficulty lies in the exploitation of the condition "f(0)⋅f(1) > 0", fully examining the depth and flexibility of mathematical thinking, making it a challenging problem.

answer:Since (x-1)^4(x+2)^8 = ax^{12} + a_1x^{11} + ldots + a_{11}x + a_{12}, When x=1, we have a + a_1 + a_2 + ldots + a_{12} = 0, (1) When x=-1, we have a - a_1 + a_2 - ldots - a_{11} + a_{12} = 16, (2) Adding equation (1) and equation (2), we get: 2(a + a_2 + a_4 + ldots + a_{12}) = 16, Therefore, a + a_2 + a_4 + ldots + a_{12} = 8; Since the coefficient of the x^{12} term is 1, i.e., a = 1, Therefore, a_2 + a_4 + ldots + a_{12} = 7. Hence, the answer is boxed{7}.

answer:To calculate Julio's total earnings over the 3 weeks, we break down the calculation into several steps: 1. **Calculate the number of customers in the second week:** - The number of customers in the first week is 35. - Since the second week has twice as many customers as the first week, we calculate this as 35 times 2 = 70 customers. 2. **Calculate the number of customers in the third week:** - The third week has triple the number of customers as the first week, so we calculate this as 35 times 3 = 105 customers. 3. **Calculate the total number of customers over the 3 weeks:** - Adding the customers from all three weeks, we get 35 + 70 + 105 = 210 customers. 4. **Calculate the total commission Julio receives:** - Julio earns a 1 commission for every customer. Therefore, for 210 customers, he earns 210 times 1 = 210. 5. **Calculate Julio's total income:** - Julio's total income includes his salary, commission, and bonus. Therefore, we add these amounts: 210 (text{commission}) + 500 (text{salary}) + 50 (text{bonus}) = 760. Thus, Julio's total earnings for the 3 weeks are boxed{760}.