Final answer:
The rational root theorem yields all possible rational zeros for the polynomial P(x)=3x³−x²+13x−18 as ±1, ±2, ±3, ±6, ±9, ±18, ±1/3, ±2/3, ±6/3, ±9/3, ±18/3, with some being duplicates after reduction. These are potential zeros and must be tested using synthetic division or another suitable method to confirm if they are actual zeros of the polynomial.
Step-by-step explanation:
The rational root theorem is a useful mathematical tool used to list all possible rational zeros of a polynomial function. Given the function P(x)=3x³−x²+13x−18, we can apply this theorem to find the potential zeros. To do this, we list the factors of the constant term (-18) and the leading coefficient (3).
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- Factors of -18: ±1, ±2, ±3, ±6, ±9, ±18
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- Factors of 3: ±1, ±3
According to the rational root theorem, the possible rational zeros are the factors of -18 divided by the factors of 3, so we have:
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- ±1, ±2, ±3, ±6, ±9, ±18
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- ±1/3, ±2/3, ±6/3, ±9/3, ±18/3
Reducing any fractions and removing duplicates, we'd get the complete list of possible rational zeros. To determine which, if any, of these are actual zeros of the polynomial, one would typically use techniques such as synthetic division or polynomial division to test these possible zeros. However, remember that not all listed potential zeros will actually be zeros of the function.