To simplify the expression \( \frac{18x^2 - 12x + 6}{3x} \), we can start by factoring out the common factor in the numerator.
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First, we'll factor the numerator \( 18x^2 - 12x + 6 \):
- Each term in \( 18x^2 - 12x + 6 \) can be divided by 6: \[ 18x^2 - 12x + 6 = 6(3x^2 - 2x + 1) \]
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Now substitute this back into the fraction: \[ \frac{18x^2 - 12x + 6}{3x} = \frac{6(3x^2 - 2x + 1)}{3x} \]
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We can further simplify by canceling: \[ = \frac{6}{3} \cdot \frac{3x^2 - 2x + 1}{x} = 2 \cdot \frac{3x^2 - 2x + 1}{x} \]
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Now, distribute the \( \frac{1}{x} \) to each term in the polynomial \( 3x^2 - 2x + 1 \): \[ = 2(3x - 2 + \frac{1}{x}) = 6x - 4 + \frac{2}{x} \]
Therefore, the simplified form of the expression is: \[ 6x - 4 + \frac{2}{x} \]
Thus, the correct answer is: B. 6x - 4 + \frac{2}{x}