To divide the polynomial \( 14x^2 + 27x - 20 \) by \( 7x - 4 \), we can use polynomial long division.
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Set up the division: \[ \frac{14x^2 + 27x - 20}{7x - 4} \]
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Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{14x^2}{7x} = 2x \]
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Multiply \( 2x \) by the entire divisor \( 7x - 4 \): \[ 2x(7x - 4) = 14x^2 - 8x \]
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Subtract this result from the original polynomial: \[ (14x^2 + 27x - 20) - (14x^2 - 8x) = 27x + 8x - 20 = 35x - 20 \]
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Repeat the process with the new polynomial \( 35x - 20 \):
- Divide the leading term: \[ \frac{35x}{7x} = 5 \]
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Multiply \( 5 \) by the divisor \( 7x - 4 \): \[ 5(7x - 4) = 35x - 20 \]
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Subtract this from \( 35x - 20 \): \[ (35x - 20) - (35x - 20) = 0 \]
Since the remainder is \( 0 \), we have: \[ \frac{14x^2 + 27x - 20}{7x - 4} = 2x + 5 \]
Thus, the final answer is: \[ 2x + 5 \]