Question

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

Answer 1

`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`