Final answer:
The expression 54u²v + 18uv² / 9u - 3v simplifies to 6u²v + 2uv² / (u - v) when factoring out common terms, ending up as option 1). However, it is critical to note that we cannot truly simplify the expression if 3uv equals 0, as this implies that u or v might be zero, and we cannot divide by zero.
Step-by-step explanation:
Given that the expression 3u v equals 0, we can simplify the expression 54u²v + 18uv² / 9u - 3v. We will first factor out common terms in the numerator and the denominator.
In the numerator, both terms have a factor of 18uv, and in the denominator, there is a common factor of 3 that we can factor out. The expression can be rewritten as follows:
(18uv)(3u + v) / 3(u - v)
We can then simplify by dividing both the numerator and the denominator by the nonzero term 18uv (as 3u v = 0 implies u or v is zero, and thus we cannot divide by zero, we assume the nonzero term here). The expression simplifies to:
(3u + v) / (u - v) after dividing by 18uv.
However, because the expression 3u v is equal to 0, this would mean that u and v cannot both be nonzero. Since we cannot divide by zero, the simplification step is not valid unless u or v is not zero. This contradiction implies that the original equation provided is not solvable under the given assumption.
But looking at the provided options, the simplification steps that ignore the zero term would give us:
6u²v + 2uv² / (u - v), which corresponds to option 1).
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