The rational root theorem states that if a polynomial equation has a rational root, it can be expressed as the quotient of a factor of the constant term over a factor of the leading coefficient.
In the given equation, the constant term is 9 and the leading coefficient is 10.
Possible rational roots can be found by taking the factors of 9 (±1, ±3, ±9) and dividing them by the factors of 10 (±1, ±2, ±5, ±10).
Possible rational roots:
±1, ±1/2, ±3, ±3/2, ±9, ±9/2
To determine if any of these possible roots are actually rational roots, we can substitute them into the equation and check if the equation equals zero.
Substituting x = 1:
10(1)³ + 19(1)² - 30(1) + 9 = 10 + 19 - 30 + 9 = 8
Since the equation does not equal zero, x = 1 is not a rational root.
We can repeat this process for the other possible rational roots to check if they are actual roots.