Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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The behavior of **polynomial functions graphs** near a repeated factor is different than what we expect from polynomial functions with terms in sequential degrees. In polynomial functions with repeated factors, the end behavior and x-intercepts will always be the same as the normal polynomial functions. However, the graph will behave differently as it approaches one of the x-intercepts.

I want to talk about graphing polynomial functions that have repeated factors. Let me show you what I mean. I've got an example here; f of x equals two times x plus one times x minus two to the third power. This is a repeated factor. You can think of x minus two as being a factor three times. And the behavior of the graph near a repeated factor is going to be a little different from what we are used to.

So I want to talk about that a little bit in a second but one thing we can say right now is the n behavior is still going to work the same way it always does. For n behavior you want to look at the leading term and the leading term is going to be two x times x cubed, two x to the fourth.

So if you were graphing a function like this, you would know that the two n's are going to go upward just like two x to the fourth does.

And you also know, what the x intercepts are going to be. We'll abbreviate x intercepts. The x intercepts are going to be negative one, zero; and two zero.

The only thing we don't really know is how the graph behaves near two zero. So I want to show you a little demonstration that would give you some intuition about what happens when we have got repeated factors.

OK. Here we are with an Andromeda sketch pad. I've got the polynomial function f of x equals x plus two to the one, x minus one to the one, and x minus three to the one.

And I can change these two exponents if I want to. Now right now, this is a simple cubic function, right. It crosses the x axis in three different places.

So let me just show you, if I increase one of these exponents do you see what happens. Now that the exponent is two the graph touches the x axis without crossing it. It kind if hits it like a quadratic. In fact, if you show the approximating polynomial, the behavior of this fourth degree polynomial is actually quite like a quadratic near x equals negative two.

And if we increase it further to three, it's going to behave like a cubic. Do you see that? See how it approaches like a cubic. It flattens out negative two and then starts increasing again.

Let's take it back to two. We can do this with both of these two intercepts. I want to take x minus one to the one, make it x minus one to the two. So now you see that I've got quadratic behavior at both negative two and at one. Let me show you the approximating polynomial.

See, here's a quadratic that approximates the behavior of my now fifth degree polynomial function. And I can increase this even more.

So, what do you learn from this? The bottom line is when you have a repeated factor like x plus two squared or x minus one cubed, the graph of the function is going to behave like a quadratic near a square factor.

And it's going to behave like a cubic near a factor that's cubed. So here's your cubic behavior, here's your quadratic. Just remember, remember the shape of y equals x cubed and remember the shape of y equals x squared and that will tell you how the function behaves near that intercept.