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]}