Question

Use long division to divide 2x3+3x2−24x−36 by x2−12 . Select the quotient.

Answer 1

To perform the long division of the polynomial \(2x^3 + 3x^2 - 24x - 36\) by \(x^2 - 12\), we follow these steps:

1. **Setup the long division**:

\[
\begin{array}{r|l}
x^2 - 12 & 2x^3 + 3x^2 - 24x - 36 \\
\end{array}
\]

2. **Divide the leading term**:

Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x^2\):

\[
\frac{2x^3}{x^2} = 2x
\]

3. **Multiply and subtract**:

Multiply \(2x\) by \(x^2 - 12\):

\[
2x \cdot (x^2 - 12) = 2x^3 - 24x
\]

Now subtract this result from the original polynomial:

\[
(2x^3 + 3x^2 - 24x - 36) - (2x^3 - 24x) = 3x^2 + 0x - 36
\]

So, the new polynomial we are working with is:

\[
3x^2 - 36
\]

4. **Repeat the process**:

Now, divide the leading term \(3x^2\) by \(x^2\):

\[
\frac{3x^2}{x^2} = 3
\]

Multiply \(3\) by \(x^2 - 12\):

\[
3 \cdot (x^2 - 12) = 3x^2 - 36
\]

Subtract this from the current polynomial:

\[
(3x^2 - 36) - (3x^2 - 36) = 0
\]

5. **Final Result**:

Since the remainder is \(0\), the division is complete. The quotient obtained from the division is:

\[
2x + 3
\]

Thus, the quotient of dividing \(2x^3 + 3x^2 - 24x - 36\) by \(x^2 - 12\) is:

\[
\boxed{2x + 3}
\]