Question

Use long division to divide 2x3+3x2−24x−36 by x2−12 . Select the quotient
2x+27−72x2−12
2x+3 2
2x+3−72x2−12 2
2x+27

Answer 1

To divide \(2x^3 + 3x^2 - 24x - 36\) by \(x^2 - 12\) using long division, we will follow the steps for polynomial long division.

1. **Divide the first term**: Divide the leading term of the dividend, \(2x^3\), by the leading term of the divisor, \(x^2\):

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

2. **Multiply the entire divisor by this result**: Multiply \(2x\) by the divisor \(x^2 - 12\):

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

3. **Subtract** this product from the original polynomial:

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

4. **Repeat the process**: Now we have \(3x^2 - 36\). Divide the leading term \(3x^2\) by the leading term of the divisor \(x^2\):

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

5. **Multiply the entire divisor**: Multiply \(3\) by the divisor \(x^2 - 12\):

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

6. **Subtract** this product from the current polynomial:

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

Since the remainder is \(0\), we are done. The quotient of the division is:

\[
2x + 3
\]

Therefore, among the options provided, the correct quotient is:

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