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Algebra

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Find all of the zeros of the polynomial function and state the multiplicity of each.

f (x) = (x^2 – 16)^2

A. – 4 with multiplicity 2 and 4 with multiplicity 2

B. – 4i with multiplicity 2 and 4i with multiplicity 2

C. 4 with multiplicity 2

D. 4 with multiplicity 4

  • Algebra -

    f(x) = (x2-16)2
    = ((x+4)(x-4))2
    = (x+4)2(x-4)2
    Can you take it from here?

  • Algebra -

    Yes this is what I got, is it correct?

    C. 4 with multiplicity 2 ?

  • Algebra -

    No, it is not the case. There are four roots for a quartic equation, so one single root with multiplicity of 2 does not suffice.

    When you have the factor (x+4)2, that implies x=-4 with multiplicity of 2.
    If you repeat the process with the factor (x-4)2, you will find the answer you need.

  • Algebra -

    Wow I am confused now. So does the answer include the i? which is

    B. – 4i with multiplicity 2 and 4i with multiplicity 2

    OR

    A. – 4 with multiplicity 2 and 4 with multiplicity 2

    I am goin to say A but I could be wrong.

  • Algebra -

    A is correct. The roots are real, so there is no i involved.
    There are two distinct roots, ±4 each with multiplicity of 2. So A is the answer.

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