According to the synthetic division below, which of the following statements are true?
Check all that apply.
\begin{tabular}{rrr}
\hline \multirow{3}{3}{} & -13 & 4 \\
& 12 & -4 \\
\hline 3 & -1 & 0
\end{tabular}
A. $(x+4)$ is a factor of $3 x^{2}-13 x+4$
B. $\left(3 x^{2}-13 x+4\right) \div(x+4)=(3 x-1)$
C. The number 4 is a root of $f(x)=3 x^{2}-13 x+4$.
D. The number -4 is a root of $f(x)=3 x^{2}-13 x+4$.
E. $(x-4)$ is a factor of $3 x^{2}-13 x+4$
F. $\left(3 x^{2}-13 x+4\right) \div(x-4)=(3 x-1)$

Step 1 ：The synthetic division table is a method used to divide a polynomial by a linear factor of the form \((x - a)\). The numbers in the top row of the table represent the coefficients of the polynomial being divided, and the number in the leftmost column represents the value of 'a' in the linear factor \((x - a)\). The numbers in the bottom row represent the coefficients of the quotient polynomial after division. If the last number in the bottom row is zero, it means that \((x - a)\) is a factor of the polynomial, and 'a' is a root of the polynomial.

Step 2 ：In this case, the top row of the table represents the polynomial \(3x^2 - 13x + 4\), and the leftmost column represents the number 3, so the linear factor is \((x - 3)\). The bottom row represents the polynomial \(3x - 1\), and the last number in the bottom row is zero, which means that \((x - 3)\) is a factor of \(3x^2 - 13x + 4\), and 3 is a root of the polynomial.

Step 3 ：Therefore, the statements A, B, C, D, E, and F are not correct, because they involve the factors \((x + 4)\), \((x - 4)\), and the roots 4 and -4, which are not consistent with the synthetic division table.

Step 4 ：\(\boxed{\text{Final Answer: None of the statements A, B, C, D, E, and F are correct.}}\)