The factored form of an equation can tell you
1. number of x-intercepts
2. where they are
Ex. y = (x-5)(x+3)(x-7)
There are three x-intercepts in this equation (the three x's in the brackets). By making y=0, we can find out where the x-intercepts are. In this case, there are three ways to do so: x=5, x=-3, and x=7.
The total number of x's in the equation can tell us how the end behaviours act and the sign of the leading coefficient tells us if it is normal or reflected. Using the same example, we can see there are three x's and therefore we know the line will act like a degree 3 function (with opposite end behaviours) and will not be reflected (positive leading coefficient).
<-- y= (x-5)(x+3)(x-7)
The order (or multiplicity) of the factor tells us how the graph interacts at that particular intercept.
Ex. Order 2 -> "bounce" off the x-intercept
Order 3 -> "inflects" through the x-intercept
Ex. y= (x-2)^3(x+5)^1
In this example, we have x^3 and x^1 which multiply together to become x^4. The overall graph will act like a quartic and will have same end behaviours.
click image to see explanation.