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When solving any polynomial equation it would be a mistake to divide both sides by anything in terms of the variable you are trying to find as you will lose a solution. Losing a repeated root is also a mistake as all roots repeated or otherwise must be used.
Let us consider the equation (y-3)(y+1)=0. We can see that y=3 or -1. If however we had divided both sides by one of the brackets eg if we had divided both sides by (y+1) we would only have the y=3 solution.
This means the rule is we must do the same correct thing to both sides when solving equations and not merely the same thing to both sides. Further evidence of this is if we multiplied both sides of an equation by zero we would end up with zero=zero but this is clearly unhelpful.
It is however correct to divide both sides of an equation by a number as no solution is being lost. If factorising the number out instead you will soon see that you will have to divide both sides by that number anyway. Therefore, factorising would be correct but a wasted step here.
