What is the LCM of #z^7-18z^6+81z^5, 5z^2-405# and #2z+18#?

1 Answer
sente
Feb 3, 2017

#10z^8-90z^7-810z^6+7290z^5#

Explanation:

Factoring each polynomial, we get

#z^7-18z^6+81z^5 = z^5(z^2-18z+81) = z^5(z-9)^2#

#5z^2-405 = 5(z^2-81) = 5(z+9)(z-9)#

#2z+18 = 2(z+9)#

As the LCM must be divisible by each of the above, it must be divisible by each factor of each polynomial. The factors which appear are: #2, 5, z, z+9, z-9#.

The greatest power of #2# which appears as a factor is #2^1#.
The greatest power of #5# which appears as a factor is #5^1#.
The greatest power of #z# which appears as a factor is #z^5#.
The greatest power of #z+9# which appears is #(z+9)^1#.
The greatest power of #z-9# which appears is #(z-9)^2#.

Multiplying these together, we get the least polynomial which is divisible by each of the original polynomials, i.e. the LCM.

#2^1xx5^1xxz^5xx(z+9)^1xx(z-9)^2 = 10z^5(z+9)(z-9)^2#

#=10z^8-90z^7-810z^6+7290z^5#

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