To divide the polynomial \( 24x^4 - 24x^3 - 18x^2 \) by \( 4x^3 + 2x^2 \) using long division, we follow these steps:
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Set Up the Long Division:
\[ \begin{array}{r|rrrrr} 4x^3 + 2x^2 & 24x^4 - 24x^3 - 18x^2 \ \end{array} \]
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Divide the Leading Terms:
Divide the leading term of the dividend \( 24x^4 \) by the leading term of the divisor \( 4x^3 \):
\[ \frac{24x^4}{4x^3} = 6x \]
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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:
\[ (24x^4 - 24x^3 - 18x^2) - (24x^4 + 12x^3) = -24x^3 - 12x^3 - 18x^2 = -36x^3 - 18x^2 \]
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Bring Down the Next Term (if any):
Since there's no term to bring down, we proceed to the next step.
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Divide Again:
Now divide the new leading term \( -36x^3 \) by \( 4x^3 \):
\[ \frac{-36x^3}{4x^3} = -9 \]
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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:
\[ \boxed{6x - 9} \]