Question

Use long division to divide the polynomial 24x^4-24^3-18^2 by 4x^3+2x^2 .write your answer in standard form

Answer 1

To perform long division on the polynomials \( 24x^4 - 24x^3 - 18x^2 \) divided by \( 4x^3 + 2x^2 \), we will go through the steps as follows:

1. **Set up the long division**: Write \( 24x^4 - 24x^3 - 18x^2 \) under the division symbol and \( 4x^3 + 2x^2 \) outside.

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

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

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

Write \( 6x \) above the division line.

3. **Multiply back**: Multiply \( 6x \) by the entire divisor \( (4x^3 + 2x^2) \):

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

4. **Subtract**: Subtract \( 24x^4 + 12x^3 \) from the original polynomial:

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

5. **Bring down the next term**: There are no more terms to bring down.

6. **Repeat the process**: Now, we divide the leading term of the new polynomial \( -36x^3 \) by the leading term of the divisor \( 4x^3 \):

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

Write \(-9\) above the division line next to \(6x\).

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

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

8. **Subtract again**: Subtract \(-36x^3 - 18x^2\) from \(-36x^3 - 18x^2\):

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

9. **Conclusion**: Since the remainder is \(0\), the division is exact, and our quotient is:

\[
6x - 9
\]

10. **Final answer in standard form**: The result of the polynomial long division is:

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