Given that 2 is a root of `x^3-6x^2+3x+10=0` , find the other roots .

Answer

To find the other roots of the polynomial equation \( x^3 - 6x^2 + 3x + 10 = 0 \), given that one root is \( x = 2 \), we can follow these steps:

### Step 1: Use Synthetic Division
Since we know that \( x = 2 \) is a root, we can use synthetic division to divide the polynomial by \( x - 2 \).

1. Write down the coefficients of the polynomial:
- Coefficients: \( 1, -6, 3, 10 \)

2. Set up synthetic division:
- Use \( 2 \) as the root.
- The setup looks like this:
```
2 | 1 -6 3 10
| 2 -8 -10
-------------------
1 -4 -5 0
```

3. Perform the synthetic division:
- Bring down the \( 1 \).
- Multiply \( 2 \) by \( 1 \) and add to \( -6 \) to get \( -4 \).
- Multiply \( 2 \) by \( -4 \) and add to \( 3 \) to get \( -5 \).
- Multiply \( 2 \) by \( -5 \) and add to \( 10 \) to get \( 0 \).

The result of the synthetic division is \( x^2 - 4x - 5 \).

### Step 2: Factor the Quadratic
Now we need to factor the quadratic \( x^2 - 4x - 5 \).

1. We look for two numbers that multiply to \( -5 \) and add to \( -4 \). The numbers are \( -5 \) and \( 1 \).

2. Thus, we can factor the quadratic:
\[ x^2 - 4x - 5 = (x - 5)(x + 1) \]

### Step 3: Solve for the Remaining Roots
Now we can set each factor equal to zero to find the remaining roots.

1. Set \( x - 5 = 0 \):
\[ x = 5 \]

2. Set \( x + 1 = 0 \):
\[ x = -1 \]

### Conclusion
The roots of the polynomial \( x^3 - 6x^2 + 3x + 10 = 0 \) are:
- \( x = 2 \) (given root)
- \( x = 5 \) (found root)
- \( x = -1 \) (found root)