Question 1080238
Note: the term "zeros" is the same as "roots"


We have this as our original function
F(x) = 5(x+7)^2 (x-7)^3


I'm going to highlight the exponents in different colors <font color=red>red</font> and <font color=blue>blue</font> so things are separated clearly.
F(x) = 5(x+7)^<font color=red>2</font> (x-7)^<font color=blue>3</font>
These exponents will be important for identifying the multiplicity.


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To find the roots, we need to set f(x) equal to zero and solve for x


F(x) = 0
5(x+7)^2 (x-7)^3 = 0
(x+7)^2 = 0 or (x-7)^3 = 0
x+7= 0 or x-7 = 0
x = -7 or x = 7


So the two distinct roots are x = -7 or x = 7


The root x = -7 has multiplicity <font color=red>2</font>. Take note of the specific color coding. This <font color=red>2</font>  is an exponent for the term (x+7) which is where the root is derived from.


Similarly, the root x = 7 has multiplicity <font color=blue>3</font> because the <font color=blue>3</font> is the exponent for (x-7). 


Extra info: an even multiplicity (such as 2) means that the graph touches the x axis and turns around. Think of a parabola like shape. See the graph below where point A is located for a visual example. An odd multiplicity root is one where the graph crosses over the x axis and keeps going (though it might turn around at some point). A visual example of this is the root where point B is located in the graph below.


Here is what the graph looks like
<img src = "https://i.imgur.com/YqsWoLX.png">
Image generated by <a href = "https://www.geogebra.org/home">GeoGebra</a> (free graphing software).
The arrows indicate that the graph continues on forever along that general curve path (but the window cuts things off).