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Sagot :
The factor that has a multiplicity of 3 is x - 2, and the polynomial has 7 roots.
The polynomial
The polynomial is given as:
[tex]P(x) = (x + 7)(x - 2)\²(x^2 - 4)(x + 1)^2[/tex]
Express 4 as 2^2. So, the polynomial becomes
[tex]P(x) = (x + 7)(x - 2)\²(x^2 - 2^2)(x + 1)^2[/tex]
Apply the difference of two squares on (x^2 - 2^2).
So, we have:
[tex]P(x) = (x + 7)(x - 2)\²(x - 2)(x + 2)(x + 1)^2[/tex]
Group the common factors
[tex]P(x) = (x + 7)(x - 2)^3(x + 2)(x + 1)^2[/tex]
Multiplicity
For a factor to have a multiplicity of 3, it means that the exponent (i.e. power) is 3.
This means that x - 2 has a multiplicity of 3.
The number of roots
Next, we add the multiplicities of each factor, to get the number of roots.
[tex]n = 1 + 3 + 1 + 2[/tex]
[tex]n = 7[/tex]
Hence, the polynomial has 7 roots
Read more about polynomials at:
https://brainly.com/question/2833285
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