The LCM of a given set of polynomials is $(x+6)(x+6)(x+2)(x-2)$. Which is equivalent to the LCM?
A. $(x+6)^2(x+2)(x-2)$
B. $(x+6)^2(x+2)^2(x-2)^2$
C. $(x+6)^2(x+2)^2(x-2)$
D. $(x+6)(x-6)(x+2)(x-2)$
Answer (2)
The equivalent expression for the given LCM of polynomials is ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) , which matches option A. By combining the repeated factors of ( x + 6 ) , we simplify the expression accurately. Thus, the answer is option A.
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Rewrite the given LCM expression by combining the repeated factors: ( x + 6 ) ( x + 6 ) ( x + 2 ) ( x − 2 ) = ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) .
Compare the rewritten expression with the given options.
Identify the matching expression.
The equivalent expression for the LCM is ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) .
Explanation
Understanding the Problem The Least Common Multiple (LCM) of a set of polynomials is given as ( x + 6 ) ( x + 6 ) ( x + 2 ) ( x − 2 ) . Our goal is to find an equivalent expression for this LCM from the given options.
Rewriting the LCM We can rewrite the given LCM expression by combining the repeated factors. Since ( x + 6 ) appears twice, we can write ( x + 6 ) ( x + 6 ) as ( x + 6 ) 2 . Therefore, the LCM can be written as ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) .
Comparing with Options Now, we compare the rewritten expression, ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) , with the given options:
Option 1: ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) Option 2: ( x + 6 ) 2 ( x + 2 ) 2 ( x − 2 ) 2 Option 3: ( x + 6 ) 2 ( x + 2 ) 2 ( x − 2 ) Option 4: ( x + 6 ) ( x − 6 ) ( x + 2 ) ( x − 2 )
We can see that Option 1 matches our rewritten expression exactly.
Final Answer Therefore, the equivalent expression for the LCM is ( x + 6 ) 2 ( x + 2 ) ( x − 2 ) .
Examples
Understanding LCMs is crucial in various fields, such as engineering and computer science. For instance, when scheduling tasks that need to be performed at regular intervals, the LCM helps determine the shortest time frame in which all tasks can be synchronized. Similarly, in electrical engineering, LCM is used to analyze waveforms and frequencies in circuits, ensuring that different components operate harmoniously. This concept also extends to everyday scenarios like planning events or managing resources, where aligning different schedules or quantities requires finding the smallest common multiple.
