Step 1 :From the table, we can see that when \(x = 1\), the value of the function \(Y1 = 24x^3 - 34x^2 + 11x - 1\) equals 0. Therefore, \(x = 1\) is a solution of the equation.
Step 2 :To verify this solution using synthetic division, we set up the synthetic division as follows: \[\begin{array}{c|cc cc} 1 & 24 & -34 & 11 & -1 \\ & & 24 & -10 & 1 \\ \hline & 24 & -10 & 1 & 0 \end{array}\]
Step 3 :The bottom row represents the coefficients of the quotient polynomial, and the last number in the bottom row is the remainder. Since the remainder is 0, \(x = 1\) is indeed a solution of the equation.
Step 4 :Now, to solve the polynomial equation, we use the quotient from the synthetic division \((24x^2 - 10x + 1 = 0)\). This is a quadratic equation, which we can solve using the quadratic formula: \[x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4*24*1}}{2*24}\]
Step 5 :Solving the above equation gives: \[x = \frac{10 \pm \sqrt{100 - 96}}{48}\] \[x = \frac{10 \pm \sqrt{4}}{48}\]
Step 6 :So the solutions are \(x = \frac{1}{4}\) and \(x = \frac{1}{2}\).
Step 7 :Therefore, the solutions of the original polynomial equation are \(\boxed{x = 1, x = \frac{1}{4}, x = \frac{1}{2}}\).