Question

Use long division to divide the polynomial 24x4−24x3−18x2
by 4x3+2x2
. Write your answer in standard form.(1 point

Answer 1

To divide the polynomial \( 24x^4 - 24x^3 - 18x^2 \) by \( 4x^3 + 2x^2 \) using long division, we follow these steps:

1. **Set Up the Long Division:**

\[
\begin{array}{r|rrrrr}
4x^3 + 2x^2 & 24x^4 - 24x^3 - 18x^2 \\
\end{array}
\]

2. **Divide the Leading Terms:**

Divide the leading term of the dividend \( 24x^4 \) by the leading term of the divisor \( 4x^3 \):

\[
\frac{24x^4}{4x^3} = 6x
\]

3. **Multiply and Subtract:**

Multiply \( 6x \) by \( 4x^3 + 2x^2 \):

\[
6x \cdot (4x^3 + 2x^2) = 24x^4 + 12x^3
\]

Now subtract this from the original polynomial:

\[
(24x^4 - 24x^3 - 18x^2) - (24x^4 + 12x^3) = -24x^3 - 12x^3 - 18x^2 = -36x^3 - 18x^2
\]

4. **Bring Down the Next Term (if any):**

Since there's no term to bring down, we proceed to the next step.

5. **Divide Again:**

Now divide the new leading term \( -36x^3 \) by \( 4x^3 \):

\[
\frac{-36x^3}{4x^3} = -9
\]

6. **Multiply and Subtract Again:**

Multiply \(-9\) by \(4x^3 + 2x^2\):

\[
-9 \cdot (4x^3 + 2x^2) = -36x^3 - 18x^2
\]

Now subtract this from the current polynomial:

\[
(-36x^3 - 18x^2) - (-36x^3 - 18x^2) = 0
\]

At this point, we have no remainder.

### Final Answer:

The quotient is:

\[
6x - 9
\]

Thus, the answer in standard form is:

\[
\boxed{6x - 9}
\]