How do you list all possible rational roots for each equation, use synthetic division to find the actual rational root, then find the remaining 2 roots for \[{x^3} - 2{x^2} + 9x - 18 = 0\]?

Hint: If we want to divide polynomials using synthetic division, you should be dividing it by a linear expression and the first number or the leading coefficient should be a 1. The divisor of the given polynomial should be of degree 1. It means that the exponent of the given variable should be 1.

Complete step by step solution:
The given equation is
\[{x^3} - 2{x^2} + 9x - 18 = 0\]
The given polynomial can be simplified using factorial method as
\[{x^2}\left( {x - 2} \right) + 9\left( {x - 2} \right) = 0\]
\[\left( {x - 2} \right) \cdot \left( {{x^2} - 9} \right) = 0\]
Hence, the factors obtained are
\[\left( {x - 2} \right)\left( {x - 3} \right)\left( {x + 3} \right) = 0\]
Hence, the roots are 2, 3 and -3.
The given polynomial, let us solve using synthetic division method as
\[
  2\left| \!{\underline {\,
  { + 1 - 2 + 9 - 18} \,}} \right. \\
  0\left| \!{\underline {\,
  { + 1 - 9} \,}} \right. \\
\]
Hence,
\[\left( {{x^3} - 2{x^2} + 9x - 18} \right) = \left( {x - 2} \right) \cdot \left( {{x^2} - 9} \right) + 0\]

Therefore, the possible root are the divisors of 18, they are
\[ \pm 1| \pm 2| \pm 3| \pm 6| \pm 9| \pm 18\]

Additional information:
Synthetic division can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1 and is generally used to find the zeros of polynomials. This method is a special case of dividing a polynomial expression by a linear factor, in which the leading coefficient should be equal to 1.
Synthetic division is mainly used to find the zeros of roots of polynomials. Such a divisor is considered as the linear factor. The coefficient of the divisor variable (say x) should be also equal to 1.

Note: The key point to find the roots of an equation using synthetic division is as we know Synthetic division is used when a polynomial is to be divided by a linear expression and the leading coefficient (first number) must be a 1, if the leading coefficient is not 1, then we need to divide by the leading coefficient to turn the leading coefficient into 1.