TITLE: Simple algebra question (find the LCM of the polynomials by factoring)
QUESTION [5 upvotes]: First of all, I am so embarrassed by this.  I am tutoring this kid in math and this question came up: Find the LCM of $y^2 - 81$ and $9 - y$, which factor into $(y + 9)(y - 9)$ and $-(y - 9)$
My answer: $-(y+9)(y-9)$ 
Her book's answer: $(y+9)(y-9)$
My question: Why is the negative left off?  Isn't it ($-1$) a factor of the second binomial?

REPLY [4 votes]: Both answers are valid. Any nonzero constant can be factored out of any polynomial.
Technically speaking, finding the $\operatorname{lcm}$ of two elements is an action done in a ring. If it exists, the $\operatorname{lcm}$ is unique up to multiplication of a unit of the ring, i.e. an invertible element.
In this case, all nonzero constants are units in a polynomial ring over a field (here $\mathbb{R}$), so multiplying by a constant still gives an $\operatorname{lcm}$. If we instead are considering these polynomials over $\mathbb{Z}$, there's still no problem since $-1$ is a unit.