Graphing polynomials Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. 

How do I sketch the graph of a polynomial?
Hint.  Basically you need to know two things.


What is the general shape of a quintic graph?
What does the graph of a polynomial look like near a double or triple root?




Edit.  For this kind of question I think it is more helpful to post hints than a complete solution.  However I am adding the following because IMO many of the answers posted here have completely missed the point of the question.

So, what is the point of graph sketching?  IMO:


  The point of graph sketching is to obtain a reasonably accurate picture of the curve with minimum work.


The "minimum work" requirement is one reason (not the only reason) why plotting lots of points is (usually) a really bad idea.  I am sure that "minimum work" is also what this question means you to do, in view of the specification that you must not plot any points except the intercepts on the axes.  What I suggest for the present example is this.


The graph is a quintic and the coefficient of $x^5$ is positive.  Since we know the general shape of a quintic we can immediately say that the graph increases from $-\infty$ to $\infty$, maybe with some turning points in between.
The factor of $(x+1)^3$ shows that the graph goes through the $x$ axis at $x=-1$, making a "cubic wriggle" as it does so.  (If you don't know what a "cubic wriggle" looks like, then you should be looking at the graph of $y=x^3$ before attempting this problem.)
The factor of $(x-2)^2$ shows that the graph touches the $x$ axis at $x=2$ and then stays on the same side of the axis.  (If this is not clear, think of the graph of $y=x^2$.)
The graph cuts the $y$ axis at $y=4$.  (I wouldn't actually have bothered with this, but the question asked for it.)


So we get the following graph.  I post a hand-drawn sketch to try to make the point of how totally irrelevant graphing software should be for this kind of problem.  The graph literally took me $15$ seconds to draw, including thinking time.


Final point: the graph clearly has a turning point somewhere between $-1$ and $2$.  We could work it out if we really needed to know it, but it would add little or nothing to the general accuracy of our graph.  Remember: minimum work!!!
Take some time to compute a number of $P(x)$ values, say every unit in range $[-10,10]$. Plot them by hand and adjust the scaling. Where it seems that the function changes behavior, try intermediate values.

You'll learn more this way.