Answer: The answer is given below.
Step-by-step explanation: We are given to find the multiplicity of the roots of the following equation:
[tex]k(x)=x(x+2)^3(x+4)^2(x-5)^4.[/tex]
We can clearly say that the roots of the above equation are 0, -2, -4 and 5.
The power of (x-0) is 1, so the multiplicity of 0 is 1.
The power of (x+2) is 3, so the multiplicity of -2 is 3.
The power of (x+4) is 2, so the multiplicity of -4 is 2.
The power of (x-5) is 4, so the multiplicity of 5 is 4.
Hence completed.
The roots of a given function are [tex]0,-2,-4,5[/tex] with multiplicity [tex]1,3,2,4[/tex] respectively.
Given:
The given function is [tex]k(x)=x(x+2)^3(x+4)^2(x-5)^2[/tex].
To find:
The multiplicity of the roots of the given function.
Explanation:
The factored form of a polynomial is:
[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]
Where, a is a constant, [tex]c_1,c_2,...,c_n[/tex] are roots with multiplicity [tex]m_1,m_2,..,m_n[/tex].
The given function can be written as:
[tex]k(x)=(x-0)^1(x-(-2))^3(x-(-4))^2(x-5)^2[/tex]
Therefore, the roots of a given function are [tex]0,-2,-4,5[/tex] with multiplicity [tex]1,3,2,4[/tex] respectively.
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