Find an Equation of the Polynomial Function - Problem 3

Let’s find the equation of one more polynomial function. Here we’re looking for the equation of a quintic function r of x and this is the graph of it. Now r of x, the first thing that we can find for r of x are its x intercepts. Which are zero and 5 and that means that you’re going to have factors of x and x minus 5. That’s only 2 factors, though and we need a 5th degree polynomial so let’s take another look.

At zero notice that we have this cubic behavior. The graph actually flattens out and it kind of looks if I cover the rest of it, it kind of looks like an x cubed graph. So this is actually going to give me x cubed. And here we’ve got behavior like a quadratic. So at x minus 5, it’s going to be squared. The only thing we need to add to this is the stretch factor because we need to make the curve actually pass through the point, (3,-12). Let’s see what a has to be to make it go through that point.

R3 is a times 3 cubed and then 3 minus 5, -2². That’s got to equal -12. What do we get here? We’ve got a times 27, times 4 equals -12. Now rather than multiply this out, I know that some of it is going to cancel so let’s observe that -12 is -3 times 4. Let’s cancel right now. Cancel the 4s and then a 3 will cancel leaving a 9 and I’ll have 9a equals -1. So a is -1/9. That’s it, just write your answer, r of x is -1/9 x cubed, x minus 5 squared.

Again remember that you’re going to have to reflect this cubic behavior in your final answer and this quadratic behavior. And also note that the end behavior is the opposite of what a quintic would normally be if the leading coefficient were positive. Normally these things end by going up to infinity but this one’s going down the negative infinity so the -1/9 makes sense.