# Simple algebra question (find the LCM of the polynomials by factoring)

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?

• gcds & lcms in polynomial rings are defined only up to unit (invertible) factors. For polynomials (over fields) they usually normalized to be monic, i.e lead coef $= 1.\,$ See here for more on such unit normalization. Nov 21, 2018 at 2:02

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.