Why will any polynomial give the same zeroes as that polynomial multiplied by any number a? Let's say I have polynomial $x^3+x^2+5=0$. Even if I multiplied this by say 6, so $6(x^3+x^2+5=0)=0$, the roots will be the same. I know this seems blindingly obvious to many people, but it isn't to me. Can someone please explain.
I know it has something to do with it being equals to 0 but I don't know why. I mean x^3+x^2+5=0 is =0, but say I took $x^6+x^2=0$. They are both =0, but they won't yield the same roots. So why would $x^3+x^2+5=0$ and $6(x^3+x^2+5=0)=0$ yield the same roots, just because they are both equal to 0?
I also know that the ratio between all the terms in the polynomial will still be equal when you multiply them, but again, I don't get why that would make the multiplied polynomial yield the same zero as it's un-multiplied version.