Find an Equation of the Polynomial Function - Problem 1

I want to find the equation of another polynomial curve. Here I’ve got the graph of a quartic, 4th degree polynomial p of x. Let me find an equation for it. P of x equals, and the fist thing I want to do is identify the x intercepts, -2, -1 and 1 and those are going to determine my factors. -2 gives me x plus 2, -1 gives me x plus 1 and 1 gives me x minus 1.

But remember when you’ve got this kind of behavior, when you’ve got the curve coming up and bouncing off the x axis, it’s behaving a little bit like a quadratic so this is going to be squared. You also know it’s squared because we have a quartic and we got to, this has to turn out to be a 4th degree polynomial.

The other thing we have to do is put a little a out in front, in order to determine what the stretch factor on this polynomial is. Let’s use this point to figure out what a is. This point is (0, -10). When I plug in zero, so p of zero, I get a times zero plus 2, 2, zero plus 1, 1, -1 squared equals -10. This is 1 times 2, 2a equals -10. And that means a is -5. That’s it, we’re done. We just need to fill out p of x equals -5, x plus 2, x plus 1, x minus 1 squared. Don’t forget to use the behavior near the intercept to determine what the power of the factor is going to be.