Answer:
The correct option is C.
Step-by-step explanation:
The given function is
\(f(x)=60x^4+86x^3-46x^2-43x+8\)
According to the rational root theorem, the potential root of the function are in the form of
\(x=\pm \frac{\text{Factors of constant term}}{\text{Factors of leading coefficient}}\)
Here, constant term is 8 and leading coefficient is 60.
Factors of 8 are ±1, ±2, ±4, ±8 and factors of 60 are ±1,±2, ±3, ±4, ±5,±6,±10,±12,±15,±20,±30,±60.
So possible roots are,
\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{5}, \pm \frac{1}{6},...\)
If f(c)=0, then (x-c) is a factor of f(x).
A. x – 6
\(f(6)=60(6)^4+86(6)^3-46(6)^2-43(6)+8=94430\)
B. 5x – 8
\(f(\frac{8}{5})=60(\frac{8}{5})^4+86(\frac{8}{5})^3-46(\frac{8}{5})^2-43(\frac{8}{5})+8=566.912\)
C. 6x – 1
\(f(\frac{1}{6})=60(\frac{1}{6})^4+86(\frac{1}{6})^3-46(\frac{1}{6})^2-43(\frac{1}{6})+8=0\)
Since the value of f(x) is 0, therefore \(\frac{1}{6}\) is a rational root and (6x-1) is a factor of given polynomial.
D. 8x + 5
\(f(\frac{-5}{8})=60(\frac{-5}{8})^4+86(\frac{-5}{8})^3-46(\frac{-5}{8})^2-43(\frac{-5}{8})+8=5.07\)
Therefore option C is correct.
