Find the least common multiple (LCM) of the polynomials [tex]$y-1$[/tex] and [tex]$1-y$[/tex].
Answer (2)
Recognize that 1 − y = − ( y − 1 ) .
Determine that the polynomials are essentially the same, differing only by a factor of − 1 .
Conclude that the LCM is y − 1 .
State the final answer: y − 1 .
Explanation
Understanding the Problem We are given two polynomials, y − 1 and 1 − y , and we need to find their least common multiple (LCM).
Rewriting the Polynomials Notice that 1 − y can be written as − ( y − 1 ) . So, we have the polynomials y − 1 and − ( y − 1 ) .
Finding the LCM The least common multiple (LCM) is the smallest expression that is a multiple of both polynomials. In this case, since 1 − y = − ( y − 1 ) , the polynomials are essentially the same, differing only by a factor of − 1 . Therefore, the LCM is simply y − 1 (or equivalently, 1 − y ).
Final Answer Thus, the least common multiple of y − 1 and 1 − y is y − 1 .
Examples
In manufacturing, if one machine produces parts with a dimension of y − 1 and another produces parts with a dimension of 1 − y , finding the LCM helps in standardizing a common dimension for compatibility. For example, if y represents a critical measurement, ensuring both machines adhere to a common multiple like y − 1 simplifies assembly and reduces waste. This concept extends to resource allocation, ensuring quantities align to avoid shortages or surpluses, thereby optimizing efficiency and minimizing costs.
The least common multiple (LCM) of the polynomials y − 1 and 1 − y is y − 1 . This is because 1 − y can be rewritten as − ( y − 1 ) , making both polynomials essentially the same. Therefore, the LCM is simply y − 1 .
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