How would you find a system of equations using a cubic polynomial?

Question

Here is the specific question

I am not even sure how to begin setting it up, I cannot understand its wording. Once it's set up, I can solve it, etc. But can someone translate what this is saying?

Answer 1

It's just saying that you have four points on the graph of the polynomial, no two of which have the same x-coordinate. Since the points are on the graph, you know that $$y_i = a + bx_i+cx_i^2+dx_i^3\tag{*}$$ for each of these points $(x_i,y_i)$.

The points are $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$. It's more convenient to refer to them collectively as $(x_i,y_i)$ for $i=1,2,3,4$.

The business with $j$ is just saying that for points with distinct subscripts, the x-coordinates are also distinct. That makes sense since otherwise $y$ wouldn't be a function of $x$ (you would violate the "vertical line" test for functions).

The system would then just be the four equations corresponding to equation (*) for the four values $i=1,2,3,4$.