SOLUTION: Which statement about the function below is true? f(x)+x^3 - 5x^2 - 25x + 125 = (x+5)(x-5)(x-5) A. The root -5 has a multiplicity of 1 B. The root -5 has a multiplicity of 2

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Question 498249: Which statement about the function below is true?
f(x)+x^3 - 5x^2 - 25x + 125 = (x+5)(x-5)(x-5)
A. The root -5 has a multiplicity of 1
B. The root -5 has a multiplicity of 2
C. The root 5 has a multiplicity of 1
D. The root 5 has a multiplicity of 3
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
your factors of the equation of:
x^3 - 5x^2 - 25x + 125 are:
(x - 5) * (x - 5) * (x + 5)
this means that the possible solutions to this equation are:
x = 5
or:
x = -5
x = 5 occurs twice because the (x - 5) factor occurs twice.
therefore x = 5 has a multiplicity of 2.
x = -5 occurs once because the (x + 5) factor occurs once.
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the answer to your question is:
A. The root -5 has a multiplicity of 1.
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here's a reference on multiplicity of roots that you might be interested in.
http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx
scroll down to the sentence that says:
"Now, we�ve got some terminology to get out of the way. If r is a zero of a polynomial and the exponent on the term that produced the root is k then we say that r has multiplicity k. Zeroes with a multiplicity of 1 are often called simple zeroes."
That's where the discussion about multiplicity of roots starts.
it says what i just told you.
the multiplicity of the root is the number of times it is a factor of the equation.
If we graph your equation of:
x^3 - 5x^2 - 25x + 125, then we will get:

you can see that the graph touches the x-axis at x = -5 and x = 5.
the graph, however, does not show you that the value of x = 5 occurs 2 times in the solution of the equation.
that comes from the examination of the factors of the equation.