For the following thinking question, please write a full solution including all work and explain your thinking process. Please note that a correct solution will not receive full marks, unless the solution is complete with all steps shown and explained. Determine the cubic function f(x) in factored form, with integral coefficients, such that it has the following properties:
a) the third differences are equal to 18
b( the remainder of f(x) divided by (x-2) is 27
c) x+1 is a factor
d) (1 ,6) is a point on f(x)

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.

if i put something wrong your not gonna like it so rather not say nothing.

we would need a picture of the quadrilateral in order to answer the question. also, it seems that the question is missing some components.