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

To determine all possible rational zeros using the Rational Zero Theorem, I start by identifying the factors involved in the polynomial’s coefficients.

The polynomial is x^3 + x^2 − 4x − 12. The leading coefficient (the coefficient of the highest-degree term) is 1, and the constant term is −12.

According to the Rational Zero Theorem, any rational zero p/q must have p as a factor of the constant term and q as a factor of the leading coefficient.

- Factors of the constant term −12 are: ±1, ±2, ±3, ±4, ±6, ±12.
- Factors of the leading coefficient 1 are: ±1 (since 1 has only 1 as a positive factor, and its sign accounts for ±).

Therefore, all possible rational zeros are the ratios p/q where p ∈ {±1, ±2, ±3, ±4, ±6, ±12} and q ∈ {±1}. Since q can only be ±1, the possible rational zeros simplify to the same integers:

Possible rational zeros: ±1, ±2, ±3, ±4, ±6, ±12.

Note on the provided answer options: in this prompt, there are no listed answer choices to evaluate against. If options were given, each option would be checked to see whether it matches one of the above potential zeros. Any option not in this set would be considered not a possible rational zero per the theorem.

If you want, we can continue by performing synthetic division or evaluating the polynomial at these candidates to identify the actual zeros among them.

ALGII141-027 4.7 Practice Problems

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

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