 # The Geometry of Polynomials

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2022-05-15 08:30:02

I recently needed to make a short demo lecture, and I thought I’d share it with you.  I’m sure I’m not the first one to notice this, but I hadn’t seen it before and I thought it was an interesting way to look at the behavior of polynomials where they cross the x-axis.

The idea is to give a geometrical meaning to an algebraic procedure:  factoring polynomials.  What is the geometry of the different factors of a polynomial?

Near the graph of the cubic looks like a parabola — and that may not be so surprising given that the factor occurs quadratically.

But which parabola, and which line?  It’s actually pretty easy to figure out.  Here is an annotated slide which illustrates the idea.

All you need to do is set aside the quadratic factor of and substitute the root, in the remaining terms of the polynomial, then simplify.  In this example, we see that the cubic behaves like the parabola near the root Note the scales on the axes; if they were the same, the parabola would have appeared much narrower.

Just isolate the linear factor substitute in the remaining terms of the polynomial, and then simplify.  Thus, the line best describes the behavior of the graph of the polynomial as it passes through the x-axis.  Again, note the scale on the axes.