The remainder obtained when `3x^(4)+7x^(3)+8x^(2)-2x-3` is divided by x+1 iss

Options

A. `-3`
B. 0
C. 3
D. 5

Explanation

To find the remainder when the polynomial \(3x^4 + 7x^3 + 8x^2 - 2x - 3\) is divided by \(x + 1\), we can use the Remainder Theorem. According to the theorem, the remainder of a polynomial \(f(x)\) when divided by \(x - c\) is \(f(c)\). In this case, since we are dividing by \(x + 1\), we will evaluate the polynomial at \(x = -1\).

### Step-by-Step Solution:

1. **Identify the polynomial**:
\[ f(x) = 3x^4 + 7x^3 + 8x^2 - 2x - 3 \]

2. **Substitute \(x = -1\)** into the polynomial:
\[ f(-1) = 3(-1)^4 + 7(-1)^3 + 8(-1)^2 - 2(-1) - 3 \]

3. **Calculate each term**:
- \(3(-1)^4 = 3 \cdot 1 = 3\)
- \(7(-1)^3 = 7 \cdot (-1) = -7\)
- \(8(-1)^2 = 8 \cdot 1 = 8\)
- \(-2(-1) = 2\)
- The constant term is \(-3\)

4. **Combine all the terms**:
\[ f(-1) = 3 - 7 + 8 + 2 - 3 \]

5. **Perform the arithmetic**:
- First, combine \(3 - 7 = -4\)
- Then, \(-4 + 8 = 4\)
- Next, \(4 + 2 = 6\)
- Finally, \(6 - 3 = 3\)

6. **Conclusion**:
The remainder when \(3x^4 + 7x^3 + 8x^2 - 2x - 3\) is divided by \(x + 1\) is \(3\).

### Final Answer:
The remainder is \(3\).

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