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

Options

A. 1
B. 10
C. 6
D. 0

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 this theorem, the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x - c\) is \(f(c)\).

### Step-by-step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \(f(x) = 3x^3 + 8x^2 - 6x + 1\).
- The divisor is \(x + 3\), which can be rewritten as \(x - (-3)\). Therefore, \(c = -3\).

2. **Evaluate the polynomial at \(c = -3\)**:
- We need to calculate \(f(-3)\):
\[ f(-3) = 3(-3)^3 + 8(-3)^2 - 6(-3) + 1 \]

3. **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 \(1\).

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

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

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

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