Use long division to divide the polynomial 24x4−24x3−18x2

Question

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:

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