MHB Help finding multiplicity and zeros?

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The expression (x+4)(x-2)^3(x^2+2x-8) can be factored to reveal its multiplicities and zeros. The zeros are -4 with a multiplicity of 2 and 2 with a multiplicity of 4. The quadratic factor simplifies to (x-2)(x+4), confirming the multiplicities. Thus, the complete factorization is (x+4)^2(x-2)^4. Understanding these factors is essential for analyzing the function's behavior.
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(x+4) (x-2)^3 (x^2+2x-8)

would it be -4 multiplicity of 2
and 2 multiplicity of 4?
 
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Yes, that's correct since the quadratic factor is the product of the other two linear factors.
 
in other words,
(x+ 4)(x- 2)^3(x^2- 2x- 8)= (x+ 4)(x- 2)^3(x- 2)(x+ 4)= (x+ 4)^2(x- 2)^4
as I am sure you realized.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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