Solve): x ^ 3 - 8x ^ 2 + 19x - 12 = 0

 To solve the equation

${𝑥}^3-8{𝑥}^2+19𝑥-12=0$, we can use different methods such as factoring, synthetic division, or the cubic formula. Let's use the Rational Root Theorem combined with synthetic division to find the roots.

1. List all possible rational roots using the Rational Root Theorem: ±(factors of constant term) / (factors of leading coefficient) Possible roots: ±1, ±2, ±3, ±4, ±6, ±12

2. Try synthetic division with each possible root until we find one that yields a remainder of zero.

Let's start with the possible roots:

1. $𝑥=1$

2. 1. Synthetic division:

      lua
        1 | 1   -8   19   -12
           |     1   -7   12
           |-----------------
              1   -7   12    0
      

   Since the remainder is zero, $𝑥=1$ is a root.

   Now we have factored our polynomial as: $(𝑥-1)({𝑥}^2-7𝑥+12)=0$

   Next, solve ${𝑥}^2-7𝑥+12=0$ using either factoring or the quadratic formula.

   ${𝑥}^2-7𝑥+12=0$

3. So the roots of the equation ${𝑥}^3-8{𝑥}^2+19𝑥-12=0$ are $𝑥=1,𝑥=3,$ and $𝑥=4$.