Esmeralda and Heinz are working to graph a polynomial function, f(x). Esmeralda says that the third-degree polynomial has four intercepts. Heinz argues that the function only crosses the x-axis three times. Is there a way for them both to be correct?

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anonymous · Answer

@ash2326 @mathmale @mathmale

anonymous · Answer

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anonymous · Answer

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mathmale · Answer

Shoot; wish I'd seen your post earlier. There are various ways in which we could approach this problem. The first thing I chose to do was to use the site wolframalpha.com. You can type formulas into the entry box and the site's programming will graph them for you. In the case of a third order polynomial, you can vary the coefficients and see how doing so will change the shape of the graph and increase or decrease the number of x-intercepts. First, I told wolframalpha that the coefficients of my third order polynomial were all 1. Take a look at the resulting graph. How many times does the graph cross the x-axis? Please see: http://www.wolframalpha.com/input/?i=a%3D1%3Bb%3D1%3Bc%3D1%3Bd%3D1%3Bax%5E3%2Bbx%5E2%2Bcx%2Bd Next, I chose to change the coefficient b to -1 from +1. Descartes' Rule of Signs hints that this may increase the number of times the graph crosses the x-axis. Not much happens! Try this yourself. Next, I chose to leave coefficients a and c positive and to change b and d to negative. What happens to the graph? Please see: http://www.wolframalpha.com/input/?i=a%3D1%3Bb%3D-1%3Bc%3D1%3Bd%3D-1%3Bax%5E3%2Bbx%5E2%2Bcx%2Bd The graph has been shifted down 2 units, but doesn't look much different.

mathmale · Answer

We could return to this approach later.  

But let's try something different now.  Keep in mind that if the coefficient a is +, the graph of any 3rd order poly will begin in Quadrant III and end in Quadrant I as x increases.  Here are various scenarios that you might encounter:

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