Question

Divide using long division
3x² - 14x - 5/ x - 5

Answer 1

To perform long division on the polynomial \( \frac{3x^2 - 14x - 5}{x - 5} \), follow these steps:

1. **Setup the long division**: Write \( 3x^2 - 14x - 5 \) under the long division symbol and \( x - 5 \) outside.

2. **Divide the first term**: Divide the leading term of the dividend \( 3x^2 \) by the leading term of the divisor \( x \):
\[
3x^2 \div x = 3x
\]

3. **Multiply and subtract**: Multiply \( 3x \) by the entire divisor \( x - 5 \):
\[
3x(x - 5) = 3x^2 - 15x
\]
Now subtract this result from the original polynomial:
\[
(3x^2 - 14x - 5) - (3x^2 - 15x) = (-14x + 15x) - 5 = x - 5
\]

4. **Bring down the next term**: The new expression is \( x - 5 \). Since there's nothing to bring down, we continue with this expression.

5. **Divide again**: Divide the leading term \( x \) by the leading term \( x \):
\[
x \div x = 1
\]

6. **Multiply and subtract**: Multiply \( 1 \) by the entire divisor \( x - 5 \):
\[
1(x - 5) = x - 5
\]
Now, subtract this from the current expression:
\[
(x - 5) - (x - 5) = 0
\]

Since the remainder is \( 0 \), the long division ends here.

The quotient of the division \( \frac{3x^2 - 14x - 5}{x - 5} \) is:
\[
\boxed{3x + 1}
\]