Question

Use long division to divide x3+4x2−6x−24 by x2−6 . Enter the coefficients of the quotient.(1 point) x+

Answer 1

To perform long division of \( x^3 + 4x^2 - 6x - 24 \) by \( x^2 - 6 \), we'll follow the standard long division process.

1. **Set up the long division**:

\[
\begin{array}{r|l}
x^2 - 6 & x^3 + 4x^2 - 6x - 24 \\
\end{array}
\]

2. **Divide the leading term**:

Divide \( x^3 \) by \( x^2 \), which gives \( x \). Write \( x \) above the division line.

3. **Multiply and subtract**:

Multiply \( x \) by \( x^2 - 6 \):

\[
x \cdot (x^2 - 6) = x^3 - 6x
\]

Now subtract this from \( x^3 + 4x^2 - 6x - 24 \):

\[
(x^3 + 4x^2 - 6x - 24) - (x^3 - 6x) = 4x^2 + 0x - 24
\]

4. **Repeat the process**:

Now we need to divide \( 4x^2 \) by \( x^2 \), which gives \( 4 \). Write \( 4 \) above the division line next to \( x \).

Multiply \( 4 \) by \( x^2 - 6 \):

\[
4 \cdot (x^2 - 6) = 4x^2 - 24
\]

Now subtract this from \( 4x^2 - 24 \):

\[
(4x^2 - 24) - (4x^2 - 24) = 0
\]

Since there are no remainder terms left, we conclude the division.

5. **Final answer**:

Thus, the quotient of \( x^3 + 4x^2 - 6x - 24 \) divided by \( x^2 - 6 \) is:

\[
x + 4
\]

The coefficients of the quotient are \( 1 \) (for \( x \)) and \( 4 \) (constant term). Hence, you can write the coefficients as:

\[
\boxed{1, 4}
\]