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Graphing Polynomial Functions with Repeated Factors - Problem 1

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 a polynomial function with repeated factors. Here is an example; graph y equals 5x² minus x³. At first we want to factor this is so that I can figure out what the x intercepts are. And there is a common factor of x² so let me pull that out.

I pull that x² and I’m left with 5 minus x. That tells me that the x intercepts are (0, 0) and (5, 0). To get the end behavior I need to look at the leading term. The leading term is the term with the highest power of x and it's minus x³. Remember the leading term may not always be first. So the end behavior is determined by minus x³ and that means it’s going to be the reflection of a cubic. It’ll look something like this. So the left tail is going to go up and the right tail is going to go down.

Now let me plot my x intercepts; (0, 0) (5, 0). And to get an idea of what the shape of this graph is, I want to plot a few points in between the intercepts. Here’s x and here’s x², 5 minus x. Let’s try, how about 2?

2² is 4, 5 minus 2, 3 so we get 12. Let me make this 16 so each of these has an increment of 4. Se here’s 12, 2, 12 goes right here. And then let’s plot, just to get an idea, I have a feeling that it’s going to come in hit the x axis, come up and then go down again. Let me just check the value at 3. 3² is 9 and 5 minus 3 is 2, so goes up to 18, so 2 past 16, quite about there.

Now let’s talk about the behavior near the intercepts. Remember when you have a repeated factor like x², the behavior near the intercept that corresponds to x² (0, 0) is going to look quadratic. It’s going to touch the x axis and just bounce off of it. Remember that the left tail is going to go up so that means that the graph is going to come in like this, bounce off the x axis, so let me draw that in, and then it will up again and it’s going to go up through this points and them come down through and then it will pass right through the x axis because this is just a simple factor, 5 minus x. Let’s see if I can do this, up through this guys and then straight down. Yeah that’s not bad. That’s our graph of y equals 5x² minus x³.

Remember when you’ve got repeated factors the behavior near the intercept that goes with that factor is not going to be simple like this, there’ll be some kind of bouncing off or maybe a cubic behavior if you have an x³ but remember intercepts and behavior, plot some points and look for the behavior near the intercepts.

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