Question 1152735
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The question is very poorly presented.  There is no answer for "THE" polynomial whose graph passes through the three points.  There is an infinite number of such polynomials.<br>
Obviously the simplest one is the constant polynomial y=4.<br>
It is easy to find other more interesting polynomials whose graphs pass through those three points.<br>
Note that the polynomial<br>
{{{y = (x-2)(x-3)(x-4)}}}<br>
passes through the points (2,0), (3,0), and (4,0).  That means the graph of<br>
{{{y = (x-2)(x-3)(x-4)+4}}}<br>
will pass through the points (2,4), (3,4), and (4,4).<br>
A graph of that third degree polynomial....<br>
{{{graph(400,400,-2,6,-2,6,(x-2)(x-3)(x-4)+4)}}}<br>
And higher degree polynomials that pass through those 4 points can be found simply by adding additional linear factors to the first part of the polynomial.  Here is a graph showing the graphs of the previous polynomial and also the fourth degree polynomial<br>
{{{y = (x-1)(x-2)(x-3)(x-4)+4}}}<br>
{{{graph(400,400,-2,6,-2,6,(x-2)(x-3)(x-4)+4,(x-1)(x-2)(x-3)(x-4)+4)}}}<br>