Question

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

Answer 1

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

1. **Write the long division setup:**

\[
\frac{24x^4 - 24x^3 - 18x^2}{4x^3 + 2x^2}
\]

2. **Divide the first term:**

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 the entire divisor \( 4x^3 + 2x^2 \):

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

Now subtract:

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

This simplifies to:

\[
-36x^3 - 18x^2
\]

4. **Repeat the process:**

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

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

5. **Multiply and subtract again:**

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

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

Now subtract:

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

6. **Result:**

Since the remainder is \( 0 \), the result of the polynomial division is:

\[
6x - 9
\]

Thus, the answer in standard form is:

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