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state the multiplicity for each root

`3(x^(2) + 5)^ {2} ( {x}^(2) - 25)^(3) (x - 8)^(4) = 0`
state the multiplicity for each root​

Answer 1

x
=
i
√
5

x
=
i
√
5
,
−
i
√
5
,
−
5
,
5
,
8

Answer 2

Answers:

• The two complex or imaginary roots
  `x = i√(5)` and
  `x = -i√(5)` have multiplicity 2.
• The two real roots x = 5 and x = -5 have multiplicity 3
• The root x = 8 has multiplicity 4

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Step-by-step explanation:

We'll use the zero product property to solve.

`3(x^2+5)^2(x^2-25)^3(x-8)^4 = 0\\\\(x^2+5)^2=0 \text{ or } (x^2-25)^3=0 \text{ or } (x-8)^4 = 0\\\\x^2+5=0 \text{ or } x^2-25=0 \text{ or } x-8 = 0\\\\x^2=-5 \text{ or } x^2=25 \text{ or } x = 8\\\\x=\pm√(-5) \text{ or } x=\pm√(25) \text{ or } x = 8\\\\x=\pm i√(5) \text{ or } x=\pm 5 \text{ or } x = 8\\\\`

where
`i = √(-1)`

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The notation
`x=\pm i√(5)` breaks up into
`x=i√(5) \text{ or } x=-i√(5)`. The multiplicity of these two roots is 2 as it's the exponent of the factor
`(x^2+5)^2`. Focus on the outermost exponent.

The notation
`x = \pm 5` becomes
`x = 5 \text{ or } x = -5`. The multiplicity of these two roots is 3 since it's the outermost exponent of the factor
`(x^2-25)^3`

And finally, the multiplicity of the root x = 8 is 4 because it is the outermost exponent of the factor
`(x-8)^4`