Answer: The answer is (B) \(\dfrac{8}{5}\) and (C) \(\dfrac{1}{6}\).
Step-by-step explanation: The given polynomial is
\(f(x)=60x^4+86x^3-46x^2-43x+8.\)
We are to select the correct option that could be a factor of the polynomial f(x) according to the Rational Root Theorem.
The Rational Root Theorem states that:
If the polynomial \(P(x)= a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0\) has any rational roots, then they must be of the form \(\pm\dfrac{\textup{factors of }a_0}{\textup{factors of }a_n}.\)
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 and factors of 8 are 1, 2, 4 and 8.
Out of the given options, only \(\dfrac{8}{5}\) and \(\dfrac{1}{6}\) can be written in the form \(\pm\dfrac{\textup{factors of }a_0}{\textup{factors of }a_n}.\), because
\(\dfrac{8}{5}=\dfrac{\textup{a factor of 8}}{\textup{a factor of 60}},\\\\\dfrac{1}{6}=\dfrac{\textup{a factor of 8}}{\textup{a factor of 60}}.\)
Thus, (B) and (C) are the correct options.
