Reconstructing a Graph From It's Intersections With the Axes

Given the intersections of a graph with the axes, we may be able to construct the equation of the graph. Suppose a quartic polynomial - degree 4 - crosses the  
  axis at  
\[x=-3, \: x=- \frac{1}{4}\]
  and touches at  
\[x= \frac{3}{2}\]
. The graph crosses the  
Each crossing of the  
  axis results in a single linear factor and each touching results in a repeated linear factor.
\[f(x)=k(x+3)(x+ \frac{1}{4})(x- \frac{3}{2})^n\]
The powers of the factors add up to 4 since  
  is a quartic, so  
  passes through the point  
\[9=k( \times 3 \times \frac{1}{4} \times (- \frac{3}{2})^2 \rightarrow k = \frac{16}{3}\]

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