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:

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