Question

divide \left(n^2+7n-34\right)\div\left(n-3\right)

Answer 1

To divide \( n^2 + 7n - 34 \) by \( n - 3 \), we can use polynomial long division.

1. **Set up the division**:
- Dividend: \( n^2 + 7n - 34 \)
- Divisor: \( n - 3 \)

2. **Divide the leading term**:
- Divide \( n^2 \) by \( n \) to get \( n \).

3. **Multiply**:
- Multiply \( n \) by \( n - 3 \) to get \( n^2 - 3n \).

4. **Subtract**:
- Subtract \( n^2 - 3n \) from \( n^2 + 7n - 34 \):
\[
(n^2 + 7n - 34) - (n^2 - 3n) = 7n + 3n - 34 = 10n - 34
\]

5. **Repeat the process**:
- Now, divide the leading term \( 10n \) by \( n \) to get \( 10 \).
- Multiply \( 10 \) by \( n - 3 \) to get \( 10n - 30 \).
- Subtract \( 10n - 30 \) from \( 10n - 34 \):
\[
(10n - 34) - (10n - 30) = -34 + 30 = -4
\]

6. **Conclusion**:
- The quotient is \( n + 10 \) and the remainder is \(-4\).
- Thus, we can write:
\[
n^2 + 7n - 34 = (n - 3)(n + 10) - 4
\]

So, the result of the division is:
\[
n + 10 + \frac{-4}{n - 3}
\]

Or in a complete form:
\[
n + 10 - \frac{4}{n - 3}
\]