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

Question

Elijah Allen · Accepted Answer

The rational root theorem helps us determine that this #f(x)# has no rational zeros, only irrational and/or Complex ones.

#f(x) = x^4-x-4# By the rational roots theorem, any rational zeros of #f(x)# must be expressible in the form #p/q# for integers #p#, #q# with #p# a divisor of the constant term #-4# and #q# a divisor of the coefficient #1# of the leading term. That means that the only possible rational zeros are: #+-1#, #+-2#, #+-4# Trying each in turn, we find: #f(1) = 1-1-4 = -4# #f(-1) = 1+1-4 = -2# #f(2) = 16-2-4 = 10# #f(-2) = 16+2-4 = 14# #f(4) = 256-4-4 = 248# #f(-4) = 256+4-4 = 256# So there are no rational zeros, but #f(x)# changes sign in #(-2, -1)# and #(1, 2)#, so there are irrational zeros in those intervals. That's as much as we can learn about this #f(x)# from the rational root theorem.

Elias Ackerman · Answer

undefined To use the Rational Roots Theorem to find possible zeros of $ f(x) = x^4 - x - 4 $, you first need to identify the leading coefficient and constant term of the polynomial. Then, you list all the factors of the constant term and divide them by all the factors of the leading coefficient. This will give you a list of possible rational roots (zeros) of the polynomial. Finally, you can use synthetic division or polynomial long division to test each potential zero to see if it produces a remainder of zero when substituted into the polynomial. Any zero that produces a remainder of zero is a root of the polynomial.