according to the rational root theorem, what are all the potential rational roots of $f(x)=15x^{11}-6x^{8}+x^…

Explanation:

Step1: Identify leading - coefficient and constant term

The polynomial is $f(x)=15x^{11}-6x^{8}+x^{3}-4x + 3$. The leading - coefficient $a_{n}=15$ and the constant term $a_{0}=3$.

Step2: Find factors of leading - coefficient and constant term

The factors of $a_{n}=15$ are $\pm1,\pm3,\pm5,\pm15$. The factors of $a_{0}=3$ are $\pm1,\pm3$.

Step3: Apply Rational Root Theorem

The Rational Root Theorem states that if a polynomial $f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}$ has integer coefficients, then the possible rational roots are of the form $\pm\frac{p}{q}$, where $p$ is a factor of the constant term $a_{0}$ and $q$ is a factor of the leading - coefficient $a_{n}$. The possible rational roots are $\pm\frac{1}{1},\pm\frac{1}{3},\pm\frac{1}{5},\pm\frac{1}{15},\pm\frac{3}{1},\pm\frac{3}{3},\pm\frac{3}{5},\pm\frac{3}{15}$. Simplifying, we get $\pm\frac{1}{15},\pm\frac{1}{5},\pm\frac{1}{3},\pm\frac{3}{5},\pm1,\pm3$.

Answer:

$\pm\frac{1}{15},\pm\frac{1}{5},\pm\frac{1}{3},\pm\frac{3}{5},\pm1,\pm3$