To perform long division on the polynomial \(-30x^6 + 48x^5 + 50x - 80\) divided by \(5x - 8\), we will follow the long division process step by step.
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Set up the division: Write \( -30x^6 + 48x^5 + 0x^4 + 50x - 80\) inside the division symbol and \(5x - 8\) outside.
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Divide the leading term: Divide \(-30x^6\) by \(5x\): \[ \frac{-30x^6}{5x} = -6x^5 \] Write \(-6x^5\) above the division bar.
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Multiply: Multiply \(-6x^5\) by \(5x - 8\): \[ -6x^5(5x - 8) = -30x^6 + 48x^5 \]
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Subtract: Subtract this product from the original polynomial: \[ (-30x^6 + 48x^5 + 0x^4 + 50x - 80) - (-30x^6 + 48x^5) = 0x^6 + 0x^5 + 0x^4 + 50x - 80 = 50x - 80 \]
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Bring down: Since there are no more higher degree terms to bring down, we proceed with \(50x - 80\).
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Repeat the process: Divide \(50x\) by \(5x\): \[ \frac{50x}{5x} = 10 \] Write \(10\) above the division bar.
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Multiply: Multiply \(10\) by \(5x - 8\): \[ 10(5x - 8) = 50x - 80 \]
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Subtract: Subtract this product from \(50x - 80\): \[ (50x - 80) - (50x - 80) = 0 \]
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Result: Since there is no remainder, the division is complete.
Collecting the results, we have:
\[ -30x^6 + 48x^5 + 50x - 80 \div (5x - 8) = -6x^5 + 10 \]
Thus, the final answer in standard form is:
\[ \boxed{-6x^5 + 10} \]