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Answer :
Final answer:
The zeros of the polynomial function f(x) = 3x^4 - 23x^3 + 7x^2 + 47x + 14 are x = -1, x = -2, x = -7, and x = 1/3.
Explanation:
To find the zeros of the polynomial function f(x) = 3x^4 - 23x^3 + 7x^2 + 47x + 14, we can use the rational root theorem to identify potential rational roots. According to the theorem, the possible rational roots are the factors of the constant term (14) divided by the factors of the leading coefficient (3).
The factors of 14 are ±1, ±2, ±7, and ±14. The factors of 3 are ±1 and ±3. Therefore, the possible rational roots are:
- ±1/1, ±2/1, ±7/1, ±14/1
- ±1/3, ±2/3, ±7/3, ±14/3
We can now test these potential roots using synthetic division or by substituting them into the polynomial function and checking if the result is zero. By testing these roots, we find that the zeros of the polynomial function are:
- x = -1
- x = -2
- x = -7
- x = 1/3
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