Quadratics - why do roots remain the same when dividing an expression?

Hi,

Could anybody explain why if I divide a quadratic, the roots remain the same? Or point me in the direction of an article or tutorial that explains this.

Take for example: -3x^2 - 15x + 9

I can divide this by 2, by (x^3 - 9x + 23), by (9x^4 - 5x^3 + 6x^2 9x + 9000)^(97/2) and so on. I see the peaks changing, yet if I'm asked to find the roots, apparently I can just discount the denominator.. for.. some reason. I've tried to find out why, but many places seem to just take it for granted without explanation. Maybe it's just really obvious (Thinking)

Thank you,

J.K