To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, follow these steps:
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Set up the long division: Write \( 15x^2 + 14x - 8 \) under the long division bar and \( 3x + 4 \) outside.
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Divide the leading term: Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \): \[ \frac{15x^2}{3x} = 5x \]
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Multiply: Multiply \( 5x \) by the entire divisor \( 3x + 4 \): \[ 5x(3x + 4) = 15x^2 + 20x \]
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Subtract: Subtract this result from the original polynomial: \[ (15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8 \]
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Bring down: There is nothing to bring down in this case.
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Repeat: Now divide the leading term \(-6x\) by \(3x\): \[ \frac{-6x}{3x} = -2 \]
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Multiply: Multiply \(-2\) by the divisor: \[ -2(3x + 4) = -6x - 8 \]
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Subtract: Subtract again: \[ (-6x - 8) - (-6x - 8) = 0 \]
At this point, there is no remainder. Thus, the total quotient of the division is: \[ \boxed{5x - 2} \]
So the final answer is: 5x−2