To simplify the expression \( 36x - 216x^2 - 36 \), we can first factor it.
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Rearranging the terms: We rewrite the expression in a standard polynomial form: \[ -216x^2 + 36x - 36. \]
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Factoring out the common factor: The greatest common factor of all the coefficients \( -216, 36, \) and \( -36 \) is \( -36 \). Therefore, we can factor out \( -36 \): \[ -36(6x - 6 - 6x^2). \]
We can rearrange it: \[ -36(-216)x^2 + 36x - 36. \]
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Now, let's simplify further by factoring \( 6 \) from the expression inside the parentheses: \[ -36(6(-36x^2 + 6x - 1)). \]
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Inverse the sign of each factor: \[ -36(6x(-36x^2 + 6x - 1)). \]
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Factoring inside the parentheses: The polynomial \( -36x^2 + 6x - 1 \) could potentially be a quadratic that can be factored, but evaluating reveals it isn't suitable for traditional factoring methods without knowing the roots. Thus, we can proceed with the process of simplifying or equating this to convert this expression normally.
Now, let's continue with an initial form and separate variables in such a way to make it clear:
- Widely, observing \( x \) results in the powered factor cancelling by \( 6x(1-x), \) or substituting for most existing ideas where variables indicate \( x^{-1} \).
Therefore, the form simplifies to: \[ 6x + \frac{6}{x} \]
Finding significant forms faster yields distance only to reveal typical examination:
- None of the listed forms seems accurately representing fractional or resultant evaluation. Nonetheless, each continuous retry yields changes towards converting \( x^2, \) reversely into operational fractions. Taking such will make \( x^{-1} \).
Thus, upon iterating confirming, whichever form avoids cancellation will remain: \[ 36x - 12 - 1 \text{ or simply constant derived.} \]
Given options can be refocused too toward \( x \) or yielding \( 6 \) uniquely present:
- The correct answer out of your choices, and assuming best-formed selections indicate:
Thus, I can summarize the final one upon clear viewing; the closest option reducing fracture forms wise should laser-focus \( 6/(x) \) as so:
Final answer in simplest form:
\[
\frac{6}{x}
\]
1 answer