# How do you use the rational roots theorem to find all possible zeros of #f(x) = x^4 + 3x^3 - 4x^2 + 5x -12#?

##### 1 Answer

#### Explanation:

Given:

#f(x) = x^4+3x^3-4x^2+5x-12#

By the rational root theorem, any rational zeros of

That means that the only possible rational zeros are:

#+-1# ,#+-2# ,#+-3# ,#+-4# ,#+-6# ,#+-12#

None of these work, so

#f(-6) = 462 > 0#

#f(-4) = -32 < 0#

#f(1) = -7 < 0#

#f(2) = 22 > 0#

So

Here's a graph of

graph{(x^4+3x^3-4x^2+5x-12)/10 [-10.705, 9.295, -6.56, 3.44]}

This particular quartic is messy to solve algebraically.

You can use a numerical method such as Durand Kerner, to find approximations to the zeros of our quartic:

#-4.3358#

#1.4332#

#-0.04871+-1.38880i#

**Footnote**

Interestingly, the similar looking quartic

#x^4+3x^3-4x^2+6x-12 = (x^2+2)(x^2+3x-6)#

Still no rational factors, but much, much easier.