Solving Quartic Equation using Derivative

[asd1249jf]

Homework Statement

$x^4 + 3x^3 - 7x^2 - 15x + 18 = 0$

Find x

Homework Equations

First Derivative?
Second Derivative?

The Attempt at a Solution

Normally, I'd use Ferrari's method to tackle this problem but I am required to use derivation to find the answer...

Where do I go from here? Do I use first derivative to find critical points? second derivative for concavity? I'm not sure what those will be helpful for though..

 

[Mark44]

Homework Statement

$x^4 + 3x^3 - 7x^2 - 15x + 18 = 0$

Find x

Homework Equations

First Derivative?
Second Derivative?

Neither. The derivatives give you information about the shape of the curve, but not about x intercepts. Try factoring. I found one root using synthetic division immediately.

The Attempt at a Solution

Normally, I'd use Ferrari's method to tackle this problem but I am required to use derivation to find the answer...

Where do I go from here? Do I use first derivative to find critical points? second derivative for concavity? I'm not sure what those will be helpful for though..

 

[Dick]

Mark44 is right, of course. The trick is to try and factor it. To help with that guessing game you should look at the rational roots theorem. Any rational root must be an integer divisor of 18.

 

[gb7nash]

And if you have no luck with the rational root theorem...

http://planetmath.org/encyclopedia/QuarticFormula.html

 

[Dick]

And if you have no luck with the rational root theorem...

http://planetmath.org/encyclopedia/QuarticFormula.html

That's the Ferrari formula i46kok was talking about in the first post. Have you EVER tried to use it in a practical problem? It's absolutely horrid. :) It may as well not even exist. And the cubic isn't much better. You were joking, right?

 

[gb7nash]

That's the Ferrari formula i46kok was talking about in the first post. Have you EVER tried to use it in a practical problem? It's absolutely horrid. :) It may as well not even exist. And the cubic isn't much better. You were joking, right?

Yeah, it's a last ditch effort When I was taking algebra, my professor made us turn it into a depressed/reduced quartic, and manipulate it and solve it as shown on this page:

http://www.sosmath.com/algebra/factor/fac12/fac12.html

Now that's horrid. But yeah, both methods are atrocious. One little arithmetic mistake and you're screwed.