Using your understanding of roots of a polynomial, explain how you would graph without a calculator f(x)=-(x+1)^2(x-3)^3

Answer 1

Clearly, our graph would have to include the two roots, -1 and 3, and go a little beyond it. Since it is a fifth degree polynomial, it will rise quickly. Since 3 is a triple root, there will be an inflection there, and since -1 is a double root, this will be a critical point. With the leading coefficient of -1, y goes to infinity as x goes to -infinity, and y goes to negative infinity as x goes to infinity. -1 will be a relative minimum. There will be a relative maximum between -1 and 3. Since 3 is a triple root and -1 is a double root, the relative maximum will have an x value a little bit greater than 1. I would probably graph x from -3 to 5.