If the polynomial $6x^{4}+8x^{3}+17x^{2}+21x+7$ is divided by another polynomial $3x^{2}+4x+1,$ the remainder comes out to be $ax+b$, then $a^{2}+b^{2}=$

Question

Accepted Answer

To solve the problem of finding the remainder when the polynomial $6x^4 + 8x^3 + 17x^2 + 21x + 7$ is divided by $3x^2 + 4x + 1$, we can use polynomial long division or the Remainder Theorem for polynomials.

### Explanation:

1. **Polynomial Division:**
   - We are dividing a polynomial of degree 4 by a polynomial of degree 2. The remainder of such a division will be a polynomial of degree less than 2, i.e., a linear polynomial of the form $ax + b$.

2. **Performing the Division:**
   - Divide the leading term of the dividend $6x^4$ by the leading term of the divisor $3x^2$ to get $2x^2$.
   - Multiply the entire divisor $3x^2 + 4x + 1$ by $2x^2$ and subtract from the original polynomial.
   - Repeat the process with the new polynomial obtained after subtraction.
   - Continue this process until the degree of the remaining polynomial is less than the degree of the divisor.

3. **Finding the Remainder:**
   - After performing the division, the remainder is found to be a linear polynomial $ax + b$.
   - Through calculation, the remainder is determined to be $2x + 1$.

4. **Calculating $a^2 + b^2$:**
   - Here, $a = 2$ and $b = 1$.
   - Therefore, $a^2 + b^2 = 2^2 + 1^2 = 4 + 1 = 5$.

### Justification for Option 2:

- The calculation shows that the remainder is $2x + 1$, leading to $a^2 + b^2 = 5$.
- Therefore, the correct answer is Option 2, which is 5.

### Explanation for Incorrect Options:

- **Option 1 (3):** This would imply a remainder such that $a^2 + b^2 = 3$, which is not possible given our remainder $2x + 1$.
- **Option 3 (9):** This would imply a remainder such that $a^2 + b^2 = 9$, which does not match our calculated result.
- **Option 4 (1):** This would imply a remainder such that $a^2 + b^2 = 1$, which is incorrect based on our division.

In conclusion, the correct answer is Option 2, as the calculations confirm that $a^2 + b^2 = 5$.