Find all possible rational zeros using the rational zero theorem.

x3+x2−4x−12

Question

Find all possible rational zeros using the rational zero theorem.

x3+x2−4x−12

BlackTom AI · Accepted Answer

Starting with the given polynomial x^3 + x^2 − 4x − 12, we apply the Rational Root Theorem to identify possible rational zeros.

First, note that the leading coefficient is 1, which means all potential rational zeros must be of the form ±(factors of the constant term  −12).

Next, list the positive and negative factors of 12: 1, 2, 3, 4, 6, 12. Therefore, the candidate rational zeros are ±1, ±2, ±3, ±4, ±6, and ±12.

This set represents all possible rational zeros according to the theorem; it does not guarantee that each candidate is actually a root. To determine which candidates (if any) are actual zeros, one would substitute each value into the polynomial or use synthetic division to check for a zero remainder.

It’s also worth noting a few important caveats: since the leading coefficient is 1, there’s no need to consider fractions with a nontrivial denominator—the q factors are just ±1, which keeps the candidate list exactly as ±1, ±2, ±3, ±4, ±6, ±12.

If you perform substitutions or testing, you’d verify whether any of these candidates satisfy x^3 + x^2 − 4x − 12 = 0. If none do, then the polynomial has no rational zeros, and its zeros are irrational or complex.

Find all possible rational zeros using the rational zero theorem. x3+x2−4x−12

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