To divide \(6x^3 - 5x^2 + 13x - 4\) by \(3x - 1\) using polynomial long division, follow these steps:
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Set up the long division: Write \(6x^3 - 5x^2 + 13x - 4\) under the long division symbol and \(3x - 1\) outside.
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Divide the leading terms: Divide the leading term of the dividend (\(6x^3\)) by the leading term of the divisor (\(3x\)):
\[ \frac{6x^3}{3x} = 2x^2 \]
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Multiply and subtract: Multiply the entire divisor \(3x - 1\) by \(2x^2\) and subtract the result from the original polynomial:
\[ 2x^2(3x - 1) = 6x^3 - 2x^2 \]
Now, perform the subtraction:
\[ (6x^3 - 5x^2 + 13x - 4) - (6x^3 - 2x^2) = -5x^2 + 2x^2 + 13x - 4 = -3x^2 + 13x - 4 \]
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Repeat the process: Now divide the leading term of the new polynomial \(-3x^2\) by the leading term of the divisor \(3x\):
\[ \frac{-3x^2}{3x} = -x \]
Multiply \(3x - 1\) by \(-x\):
\[ -x(3x - 1) = -3x^2 + x \]
Now, subtract this from \(-3x^2 + 13x - 4\):
\[ (-3x^2 + 13x - 4) - (-3x^2 + x) = 13x - x - 4 = 12x - 4 \]
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Repeat again: Now divide the leading term \(12x\) by the leading term \(3x\):
\[ \frac{12x}{3x} = 4 \]
Multiply \(3x - 1\) by \(4\):
\[ 4(3x - 1) = 12x - 4 \]
Now, perform the subtraction:
\[ (12x - 4) - (12x - 4) = 0 \]
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Conclusion: Since the remainder is \(0\), we can conclude that:
\[ \frac{6x^3 - 5x^2 + 13x - 4}{3x - 1} = 2x^2 - x + 4 \]
Thus, the result of the polynomial long division is:
\[ 2x^2 - x + 4 \]
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