Find the least common multiple (LCM) of the polynomials [tex]$z^2-9$[/tex] and [tex]$7 z+21$[/tex].
Answer (2)
Factor the polynomials: z 2 − 9 = ( z − 3 ) ( z + 3 ) and 7 z + 21 = 7 ( z + 3 ) .
Identify the distinct factors: 7, ( z − 3 ) , and ( z + 3 ) .
Multiply the highest power of each distinct factor to find the LCM: 7 ( z − 3 ) ( z + 3 ) .
Expand the expression to obtain the LCM in polynomial form: 7 z 2 − 63 .
Explanation
Understanding the Problem We are asked to find the least common multiple (LCM) of the polynomials z 2 − 9 and 7 z + 21 . The LCM is the smallest polynomial that is divisible by both given polynomials.
Factoring the Polynomials First, we factor each polynomial completely. The first polynomial is a difference of squares: z 2 − 9 = ( z − 3 ) ( z + 3 ) The second polynomial can be factored by taking out the common factor 7: 7 z + 21 = 7 ( z + 3 )
Finding the LCM To find the LCM, we take the highest power of each distinct factor present in the factorizations. The distinct factors are 7, ( z − 3 ) , and ( z + 3 ) . The highest power of each factor is 1. Therefore, the LCM is: L CM = 7 ( z − 3 ) ( z + 3 )
Expanding the LCM Now, we expand the expression to obtain the LCM in polynomial form: 7 ( z − 3 ) ( z + 3 ) = 7 ( z 2 − 9 ) = 7 z 2 − 63 Thus, the least common multiple of z 2 − 9 and 7 z + 21 is 7 z 2 − 63 .
Examples
The concept of LCM is used in various real-life scenarios, such as scheduling events. For instance, if one event occurs every z 2 − 9 days and another event occurs every 7 z + 21 days, the LCM helps determine when both events will occur on the same day. In this case, the LCM 7 z 2 − 63 represents the number of days until both events coincide. This is useful for planning and coordinating activities that occur at different intervals.
The least common multiple (LCM) of the polynomials z 2 − 9 and 7 z + 21 is 7 z 2 − 63 after factoring and identifying the distinct factors, which are multiplied together. This involves recognizing the factorizations and combining the highest powers of each factor. The expanded form of the LCM is 7 z 2 − 63 .
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