To determine how many roots of the polynomial function [tex]\( f(x) = 2x^3 - 19x^2 + 57x - 54 \)[/tex] are rational, we will analyze the roots of the equation. Here is the step-by-step process:
1. Set the function equal to zero to find its roots:
[tex]\[
2x^3 - 19x^2 + 57x - 54 = 0
\][/tex]
2. Identify the roots: We found that the roots of this polynomial equation are:
[tex]\[
2, \quad 3, \quad \text{and} \quad \frac{9}{2}
\][/tex]
3. Check the rationality of each root:
- The root [tex]\( 2 \)[/tex] is an integer, hence it is a rational number.
- The root [tex]\( 3 \)[/tex] is also an integer, hence it is a rational number.
- The root [tex]\( \frac{9}{2} \)[/tex] is a rational number because it is a ratio of two integers.
4. Count the number of rational roots: We observe that:
- Root [tex]\( 2 \)[/tex] is rational.
- Root [tex]\( 3 \)[/tex] is rational.
- Root [tex]\( \frac{9}{2} \)[/tex] is rational but expressed as a fraction.
Thus, there is only one root expressed in fractional form, which is [tex]\( \frac{9}{2} \)[/tex].
Hence, the number of rational roots of [tex]\( f(x) \)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
