To perform polynomial long division with \(6x^3 - 5x^2 + 13x - 4\) divided by \(3x - 1\), we follow these steps:
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Set up the division: Write the dividend \(6x^3 - 5x^2 + 13x - 4\) under the long division symbol and the divisor \(3x - 1\) to the left.
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Perform the division step by step:
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Step 1: Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{6x^3}{3x} = 2x^2 \]
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Step 2: Multiply \(2x^2\) by the entire divisor \(3x - 1\): \[ 2x^2(3x - 1) = 6x^3 - 2x^2 \]
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Step 3: Subtract this result from the original polynomial: \[ (6x^3 - 5x^2 + 13x - 4) - (6x^3 - 2x^2) = -5x^2 + 2x^2 + 13x - 4 = -3x^2 + 13x - 4 \]
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Step 4: Repeat the process with the new polynomial \(-3x^2 + 13x - 4\): Divide \(-3x^2\) by \(3x\): \[ \frac{-3x^2}{3x} = -x \]
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Step 5: Multiply \(-x\) by the divisor \(3x - 1\): \[ -x(3x - 1) = -3x^2 + x \]
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Step 6: Subtract this from \(-3x^2 + 13x - 4\): \[ (-3x^2 + 13x - 4) - (-3x^2 + x) = 13x - 4 - x = 12x - 4 \]
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Step 7: Repeat the process again with the new polynomial \(12x - 4\): Divide \(12x\) by \(3x\): \[ \frac{12x}{3x} = 4 \]
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Step 8: Multiply \(4\) by the divisor \(3x - 1\): \[ 4(3x - 1) = 12x - 4 \]
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Step 9: Subtract this from \(12x - 4\): \[ (12x - 4) - (12x - 4) = 0 \]
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Conclusion: Because the final remainder is 0, we conclude that: \[ \frac{6x^3 - 5x^2 + 13x - 4}{3x - 1} = 2x^2 - x + 4 \]
Thus, the result of the polynomial long division is: \[ \boxed{2x^2 - x + 4} \]
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