Question 1080238: Find the zeros of the polynomial function and state the multiplicity of each
F(x) = 5(x+7)^2 (x-7)^3 Found 2 solutions by ikleyn, jim_thompson5910:Answer by ikleyn(52817) (Show Source):
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 red and blue so things are separated clearly.
F(x) = 5(x+7)^2 (x-7)^3
These exponents will be important for identifying the multiplicity.
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 2. Take note of the specific color coding. This 2 is an exponent for the term (x+7) which is where the root is derived from.
Similarly, the root x = 7 has multiplicity 3 because the 3 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
Image generated by GeoGebra (free graphing software).
The arrows indicate that the graph continues on forever along that general curve path (but the window cuts things off).
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