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All potential rational roots of f(x) = 15x¹¹ - 6x⁸ + x³ - 4x + 3, going by the rational rots theorem are {±1, ±1/3, ±1/5, ±1/15, ±3, ±3/5}.
The rational root theorem suggests that for a polynomial function
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x¹ + a₀, all potential rational roots can be given by the formula p/q, where p is the set of all factors of ±a₀ and q is the set of all factors of ±aₙ.
In the question, we are asked to find the potential rational roots of
f(x) = 15x¹¹ - 6x⁸ + x³ - 4x + 3.
Comparing the given polynomial function with the standard polynomial function f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x¹ + a₀, we can say aₙ = 15, and a₀ = 3.
By rational root theorem, we know that all potential rational roots can be given by the formula p/q, where p is the set of all factors of ±a₀ and q is the set of all factors of ±aₙ.
Therefore, we can say that p = {factors of ±3} and q = {factors of ±15},
or, p = {±1, ±3}, q = {±1, ±3, ±5, ±15}.
Now, we can write all potential roots by calculating p/q.
p/q = {±1, ±1/3, ±1/5, ±1/15, ±3, ±3/5}
Therefore, all potential rational roots of f(x) = 15x¹¹ - 6x⁸ + x³ - 4x + 3, going by the rational roots theorem are {±1, ±1/3, ±1/5, ±1/15, ±3, ±3/5}.
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