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

Feb 18, 2015

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 $x \to \pm \infty$ the function goes to $\infty$ because of the ${x}^{4}$ dependence.

Then I evaluated the Derivatives:

First Derivative: $12 {x}^{3} - 15 {x}^{2} + 2 x - 5$
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 $\infty$ I deduced that is a minimum).
I used the following to solve the cubic:

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

Second Derivative: $36 {x}^{2} - 30 x + 2$
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 ${y}_{2} = - 6.41641$