Find the remainder when the expression `3x^(3)+8x^(2)-6x+1` is divided by `(x+3)`.

Answer

To find the remainder when the polynomial \( 3x^3 + 8x^2 - 6x + 1 \) is divided by \( x + 3 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \). In our case, we need to evaluate the polynomial at \( x = -3 \) (since we are dividing by \( x + 3 \)).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
The polynomial is \( f(x) = 3x^3 + 8x^2 - 6x + 1 \) and the divisor is \( x + 3 \).

2. **Use the Remainder Theorem**:
To find the remainder, we need to evaluate \( f(-3) \).

3. **Substitute \( x = -3 \) into the polynomial**:
\[ f(-3) = 3(-3)^3 + 8(-3)^2 - 6(-3) + 1 \]

4. **Calculate each term**:
- Calculate \( (-3)^3 \):
\[ (-3)^3 = -27 \quad \Rightarrow \quad 3(-27) = -81 \]
- Calculate \( (-3)^2 \):
\[ (-3)^2 = 9 \quad \Rightarrow \quad 8(9) = 72 \]
- Calculate \( -6(-3) \):
\[ -6(-3) = 18 \]
- The constant term is \( 1 \).

5. **Combine all the terms**:
\[ f(-3) = -81 + 72 + 18 + 1 \]

6. **Perform the addition**:
- First, combine \( -81 + 72 \):
\[ -81 + 72 = -9 \]
- Then add \( 18 \):
\[ -9 + 18 = 9 \]
- Finally, add \( 1 \):
\[ 9 + 1 = 10 \]

7. **Conclusion**:
The remainder when \( 3x^3 + 8x^2 - 6x + 1 \) is divided by \( x + 3 \) is \( 10 \).

### Final Answer:
The remainder is \( 10 \).