To simplify the expression \(\frac{4x^2 - 12x + 24}{4x}\), we can follow these steps:
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Factor the numerator: The numerator is \(4x^2 - 12x + 24\). We can factor out a 4:
\[ 4(x^2 - 3x + 6) \]
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Rewrite the expression: Now, substitute the factored numerator into the original expression:
\[ \frac{4(x^2 - 3x + 6)}{4x} \]
The 4's in the numerator and denominator cancel out:
\[ \frac{x^2 - 3x + 6}{x} \]
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Separate the expression: Now, we can split the single fraction into separate terms:
\[ \frac{x^2}{x} - \frac{3x}{x} + \frac{6}{x} \]
This simplifies to:
\[ x - 3 + \frac{6}{x} \]
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Write in standard form: The standard form of a polynomial or expression typically lists terms in descending order:
\[ x - 3 + \frac{6}{x} \]
Therefore, the final simplified expression is:
\[ x - 3 + \frac{6}{x} \]