How do you use the rational roots theorem to find all possible zeros of f(x)=x4−x−4?

1 Answer
George C.
Mar 23, 2016

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

Explanation:

f(x)=x4−x−4

By the rational roots theorem, any rational zeros of f(x) must be expressible in the form pq 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.

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