Question

How do you find the LCM of `x^8- 12x^7 +36x^6, 3x^2- 108` and `7x+ 42`?

Answer 1

The LCM is:

`21x^9-126x^8-756x^7+4536x^6`

Explanation:

For this question, it's probably easiest to factor all of the polynomials first:

`x^8-12x^7+36x^6 = x^6(x^2-12x+36) = x^6(x-6)^2`

`3x^2-108 = 3(x^2-36) = 3(x-6)(x+6)`

`7x+42 = 7(x+6)`

So the LCM of the scalar factors is that of `1`, `3` and `7`, which is `21`

The simplest product of polynomial factors including all of the linear factors we have found, in their multiplicities is:

`x^6(x-6)^2(x+6) = x^6(x-6)(x^2-36)`

`color(white)(x^6(x-6)^2(x+6)) = x^6(x^3-6x^2-36x+216)`

`color(white)(x^6(x-6)^2(x+6)) = x^9-6x^8-36x^7+216x^6`

So to get the LCM of the original polynomials, we just need to multiply this by `21`:

`21(x^9-6x^8-36x^7+216x^6) = 21x^9-126x^8-756x^7+4536x^6`