Given:
The function is
[tex]f(x)=3x^3+2x^2+x+4[/tex]
To find:
The value from the given options that is NOT a possible root of the function.
Solution:
According to rational root theorem, all possible roots are in the form of
[tex]x=\dfrac{\text{Factors of constant term}}{\text{Factors of leading coefficient}}[/tex]
We have,
[tex]f(x)=3x^3+2x^2+x+4[/tex]
Here,
Leading coefficient = 3.
Constant term = 4
Factors of 3 are ±1, ±3.
Factors of 4 are ±1, ±2, ±4.
Using rational root theorem, all possible roots of given functions are
[tex]\pm 1,\pm 2,\pm 4,\pm \dfrac{1}{3}, \pm \dfrac{2}{3},\pm \dfrac{4}{3}[/tex]
Clearly, [tex]\dfrac{4}{3},1,2[/tex] are possible roots but [tex]\dfrac{3}{4}[/tex] is not the possible root.
Therefore, the correct option is C.
