Question

How do you graph `3x^4-5x^3+x^2-5x-2` by finding all of its roots?

Answer 1

Well, this one is quite tough...or at least for me it is!
I started using Ruffini's Method to reduce the degree of the equation and find the roots:

Where the roots are my intercepts with the `x` axis (exclude the immaginary ones).
The `y` axis intercept is at `y=-2` (after setting `x=0` in your equation).

When `x->+-oo` the function goes to `oo` because of the `x^4` dependence.

Then I evaluated the Derivatives:

First Derivative: `12x^3-15x^2+2x-5`
Setting this one equal to zero should give me the points of minimum/maximum of my function.
This is not an easy task but using the cubic formula I got that:
`x=1.35415` and `y=-9.26509` which are the coordinates of the minimum of your function (considering the intercepts and the tendency at `oo` I deduced that is a minimum).
I used the following to solve the cubic:

(Reference: http://en.wikipedia.org/wiki/Cubic_function)

Second Derivative: `36x^2-30x+2`
Setting this one equal to zero should give me the points of inflection of my function. These are for:
`x_1=0.073` and `y_1=-2.36153`
`x_2=0.76` and `y_2=-6.41641`

Finally your graph: