For this case we have the following expression:
[tex]k (x) = x (x + 2) ^ 3 (x + 4) ^ 2 (x-5) ^ 4\\[/tex]
The roots are:
[tex]x = -2\\\\x = -4\\\\x = 5\\\\x = 0\\[/tex]
For example: For x = 0 we have
[tex]k (0) = 0 (0 + 2) ^ 3 (0 + 4) ^ 2 (0-5) ^ 4\\\\k (0) = 0 (2) ^ 3 (4) ^ 2 (-5) ^ 4\\\\k (0) = 0 (8) (16) (625)\\\\k (0) = 0\\[/tex]
so it is shown that [tex]x = 0[/tex] is a root.
By definition, multiplicity represents the number of times a root is repeated in a polynomial, in turn it is given by the degree of the term that contains the root.
Thus:
The multiplicity of 0 is 1
The multiplicity of -2 is 3
The multiplicity of -4 is 2
The multiplicity of 5 is 4
Answer:
The multiplicity of 0 is 1
The multiplicity of -2 is 3
The multiplicity of -4 is 2
The multiplicity of 5 is 4
