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Graphing Polynomial Functions with Repeated Factors - Problem 3

Teacher/Instructor Norm Prokup
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.

I want to graph one really hard example of a polynomial that’s got repeated factors. Here I’ve got a 5th degree polynomial which is called a quintic. First thing I want to do is identify the x intercepts. The one that comes from these factors is going to be (2, 0) and the one that comes from this factor is (-2, 0). But the behavior of the intercepts is going to be different because we have different powers here. We’ll see that in a second.

The end behavior comes from the leading term. I have to multiply to figure out what the leading term is. I have -1/3, x², x³, -1/3 x to the 5th. Now a 5th degree polynomial has a behavior kind of like a 3rd degree where the ends go in opposite directions. Normally the right end goes up and the left end goes down but with this negative, it’s going to be the reverse of that. The left end will go up and the right will go down. We’ll use that in a second.

Let’s figure out what happens between the 2 intercepts. I can plot those now. We have (-2, 0) and (2, 0). I’ll plot a few points. Remember at -2, this graph is going to behave like a cubic, so it’s going to do something like this and at 2, it’s going to behave like a quadratic. Now since it's below the x axis, it’s probably going to bounce off like that.

Let’s plot some points to be sure though. -1/3 (x minus 2)², (x plus 2)³. How about, let’s plot zero for sure. I want to know that the y intercept is. So when I plug in zero I get -1/3, -2², 2³. This is going to give me 4, 8 so I have 32, -32 over 3. This is almost -11. So, why don’t I make my scale, let’s make this 4, 8, 10, sorry 12, this is 10, 11 it’s going to be right around here. Let me just write that down, 12, -12.

Let’s plot -1, -1/3, we get -1 plus -2, -3 squared, -1 plus 2, 1 cubed so we get 9, divide by 3, -3. So at -1 we have -3. One short of this, right up there. How about positive 1? 1 minus 2, -1 squared, 1 plus 2, 3, cubed -1² is 1, 3³, 27 divide by 3 is 9, -9. Okay, so this is 8, so one past 8 is about, well this is 10, so it’s about here. I think I’ve got enough to draw a decent graph here.

Remember cubic behavior here at -2. Let’s draw this end going like that and this one going something like this. Then as we come in to this guy; remember we’re going to have quadratic behavior. We’re going to hit the x axis and bounce off. So this is a little tricky. I’m going to start from here maybe. There, that’s not too bad.

Now remember until we actually learn calculus we can’t always find the exact location of turning points like the one that happens in here somewhere but a qualitative graph like this is going to be good enough in pre-calculus. Make sure you’ve got your intercepts and the correct behavior at the intercepts, when it’s easy to do get the y intercept as well. Make sure you’ve got correct end behavior. And that’s enough to get a pretty good graph of a polynomial function with repeated factors.

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