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please solve with steps explaining step by step

We are given the cubic equation:

\[
6x^3 - 11x^2 + 6x - 1 = 0
\]
and told that one of the solutions is \(x = 1\).

### Step 1: Verify the known solution

We can substitute \(x = 1\) into the equation to verify that it satisfies the equation.

\[
6(1)^3 - 11(1)^2 + 6(1) - 1 = 6 - 11 + 6 - 1 = 0
\]

Since the equation equals zero, \(x = 1\) is indeed a solution.

### Step 2: Perform synthetic division

Since \(x = 1\) is a root, \(x - 1\) must be a factor of the cubic polynomial. To find the remaining quadratic factor, we can use synthetic division to divide the cubic polynomial by \(x - 1\).

The coefficients of the cubic polynomial \(6x^3 - 11x^2 + 6x - 1\) are: \(6, -11, 6, -1\).

Using synthetic division:

\[
\begin{array}{r|rrrr}
1 & 6  & -11 & 6  & -1 \\
  &    & 6   & -5  & 1 \\
\hline
  & 6  & -5  & 1  & 0 \\
\end{array}
\]

The quotient is \(6x^2 - 5x + 1\), and the remainder is 0, confirming that the division is exact.

Thus, we have factored the cubic equation as:

\[
6x^3 - 11x^2 + 6x - 1 = (x - 1)(6x^2 - 5x + 1)
\]

### Step 3: Solve the quadratic equation

Now, we need to solve the quadratic equation \(6x^2 - 5x + 1 = 0\). We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(6x^2 - 5x + 1 = 0\), the coefficients are \(a = 6\), \(b = -5\), and \(c = 1\). Substituting these values into the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(6)(1)}}{2(6)}
\]
\[
x = \frac{5 \pm \sqrt{25 - 24}}{12}
\]
\[
x = \frac{5 \pm \sqrt{1}}{12}
\]
\[
x = \frac{5 \pm 1}{12}
\]

So, the two solutions are:

\[
x = \frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2}
\]
\[
x = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3}
\]

### Step 4: Final solutions

The solutions to the cubic equation \(6x^3 - 11x^2 + 6x - 1 = 0\) are:

\[
x = 1, \quad x = \frac{1}{2}, \quad x = \frac{1}{3}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you perform synthetic division in detail?
2. What are the steps of the quadratic formula?
3. How do you verify if a root is valid in polynomial equations?
4. How do we factor cubic equations?
5. Can all cubic equations be factored using synthetic division?

### Tip:
Always check known roots by substitution to avoid unnecessary work when solving polynomials.

6x^3 - 11x^2 + 6x - 1 = 0, knowing that one of the solutions is 1. Please solve with steps explaining step by step.

Step-by-step solution to solving the cubic equation 6x^3 - 11x^2 + 6x - 1 = 0, knowing one solution is 1. Learn about synthetic division and the quadratic formula, ideal for high school students.

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