If one root of the equation x^3 - (k+1)x + k = 0 is 2, then the remaining roots are

We are given that one root of the cubic equation \(x^3 - (k+1)x + k = 0\) is 2, and we are tasked with finding the remaining roots.

### Step 1: Use the fact that 2 is a root.
Since 2 is a root, we can substitute \(x = 2\) into the equation:

\[
2^3 - (k+1) \cdot 2 + k = 0
\]

Simplifying this:

\[
8 - 2(k + 1) + k = 0
\]

\[
8 - 2k - 2 + k = 0
\]

\[
6 - k = 0
\]

Thus, we find that \(k = 6\).

### Step 2: Substitute \(k = 6\) into the cubic equation.
Now, substitute \(k = 6\) back into the original equation:

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

This simplifies to:

\[
x^3 - 7x + 6 = 0
\]

### Step 3: Factor the cubic equation.
We already know that \(x = 2\) is a root, so we can perform synthetic or polynomial division of \(x^3 - 7x + 6\) by \(x - 2\).

Using synthetic division:

\[
\begin{array}{r|rrrr}
2 & 1 & 0 & -7 & 6 \\
  &   & 2 & 4  & -6 \\
\hline
  & 1 & 2 & -3 & 0 \\
\end{array}
\]

The quotient is \(x^2 + 2x - 3\), and the remainder is 0, confirming that the division was correct.

### Step 4: Solve the quadratic equation.
We now solve the quadratic equation:

\[
x^2 + 2x - 3 = 0
\]

Factorizing:

\[
(x + 3)(x - 1) = 0
\]

Thus, the remaining roots are \(x = -3\) and \(x = 1\).

### Final Answer:
The remaining roots of the equation are \(x = -3\) and \(x = 1\).

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Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. How do you perform synthetic division for a cubic polynomial?
2. What is the relationship between the coefficients and roots of a cubic equation?
3. How can you find the discriminant of a cubic equation?
4. What are the general methods for solving cubic equations?
5. How can factoring help in solving polynomial equations?

### Tip:
Always check your roots by substituting them back into the original equation to ensure they satisfy it.

Learn how to solve the cubic equation x^3 - (k+1)x + k = 0, where one root is given as 2. The step-by-step solution includes using synthetic division and factoring to find the remaining roots. Ideal for high school students in Grades 10-12.

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