To divide \( 8x^3 - 20x^2 + 18x - 11 \) by \( 2x^2 - 3 \) using long division, follow these steps:
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Set up the long division:
\[ \begin{array}{r|l} 2x^2 - 3 & 8x^3 - 20x^2 + 18x - 11 \ \end{array} \]
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Divide the leading term of the dividend \( 8x^3 \) by the leading term of the divisor \( 2x^2 \) to get \( 4x \) (since \( 8x^3 \div 2x^2 = 4x \)).
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Multiply \( 4x \) by the entire divisor \( 2x^2 - 3 \):
\[ 4x(2x^2 - 3) = 8x^3 - 12x \]
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Subtract this result from the original polynomial:
\[ (8x^3 - 20x^2 + 18x - 11) - (8x^3 - 12x) = -20x^2 + 18x + 12x - 11 = -20x^2 + 30x - 11 \]
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Now, repeat the process. Divide the leading term of the new polynomial \( -20x^2 \) by the leading term of the divisor \( 2x^2 \):
\[ -20x^2 \div 2x^2 = -10 \]
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Multiply \( -10 \) by the divisor \( 2x^2 - 3 \):
\[ -10(2x^2 - 3) = -20x^2 + 30 \]
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Subtract this from \( -20x^2 + 30x - 11 \):
\[ (-20x^2 + 30x - 11) - (-20x^2 + 30) = 30x - 11 - 30 = 30x - 41 \]
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At this point, we have:
\[ \begin{array}{r|l} 2x^2 - 3 & 8x^3 - 20x^2 + 18x - 11 \ & \underline{4x} \ & -20x^2 + 30x - 11 \ & \underline{-10} \ & 30x - 41 \ \end{array} \]
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Now we can't divide anymore because the degree of the remainder \( 30x - 41 \) is less than the degree of the divisor \( 2x^2 - 3 \).
So, the final result of the division is:
\[ \text{Quotient: } 4x - 10 \] \[ \text{Remainder: } 30x - 41 \]
Therefore, the overall result can be written as:
\[ 8x^3 - 20x^2 + 18x - 11 = (2x^2 - 3)(4x - 10) + (30x - 41) \]
In conclusion, the result of dividing \( 8x^3 - 20x^2 + 18x - 11 \) by \( 2x^2 - 3 \) is:
\[ \text{Quotient: } 4x - 10, \quad \text{Remainder: } 30x - 41 \]