# How do you use the rational root theorem to find the roots of #f(x)=x^4-x-4#?

##### 1 Answer

You can't.

#### Explanation:

By the rational root theorem, any rational root of

So the only possible rational roots are:

#+-1# ,#+-2# ,#+-4#

Then we find:

#f(-4) = 256+4-4 = 256#

#f(-2) = 16+2-4 = 14#

#f(-1) = 1+1-4 = -2#

#f(1) = 1-1-4 = -4#

#f(2) = 16-2-4 = 10#

#f(4) = 256-4-4 = 248#

So

Since

Newton's method can then be used to find approximations for the two Real roots. Starting with an approximation

#a_(i+1) = a_i - f(x)/(f'(x))#

#f'(x) = 4x^3-1#

Starting with

#a_1 = -1.3232758621#

#a_2 = -1.2853460646#

#a_3 = -1.2837842190#

#a_4 = -1.2837816659#

#a_5 = -1.2837816659#

Starting with

#a_1 = 1.535#

#a_2 = 1.5337528051#

#a_3 = 1.5337511688#

#a_4 = 1.5337511688#

graph{x^4-x-4 [-10, 10, -5, 5]}