Find the remainder when the expression `3x^3+6x^2-7x+6` is divided by `x +3`.

Options

A. 1
B. 0
C. 5
D. 6

Answer

To find the remainder when the polynomial \(3x^3 + 6x^2 - 7x + 6\) 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 \(f(c)\). In this case, we will 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 \(f(x) = 3x^3 + 6x^2 - 7x + 6\).
- The divisor is \(x + 3\), which means we will evaluate \(f(-3)\).

2. **Substitute \(-3\) into the polynomial**:
\[ f(-3) = 3(-3)^3 + 6(-3)^2 - 7(-3) + 6 \]

3. **Calculate each term**:
- Calculate \((-3)^3\):
\[ (-3)^3 = -27 \quad \Rightarrow \quad 3(-27) = -81 \]
- Calculate \((-3)^2\):
\[ (-3)^2 = 9 \quad \Rightarrow \quad 6(9) = 54 \]
- Calculate \(-7(-3)\):
\[ -7(-3) = 21 \]
- The constant term is \(6\).

4. **Combine all the calculated values**:
\[ f(-3) = -81 + 54 + 21 + 6 \]

5. **Perform the addition**:
- First, combine \(-81\) and \(54\):
\[ -81 + 54 = -27 \]
- Next, add \(21\):
\[ -27 + 21 = -6 \]
- Finally, add \(6\):
\[ -6 + 6 = 0 \]

6. **Conclusion**:
- The remainder when \(3x^3 + 6x^2 - 7x + 6\) is divided by \(x + 3\) is \(0\).

### Final Answer:
The remainder is \(0\).