Find a polynomial function of degree 7 with -2 as a zero of multiplicity​ 3, 0 as a zero of multiplicity3 ​, and 2 as a zero of multiplicity

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Find a polynomial function of degree 7 with -2 as a zero of multiplicity​ 3, 0 as a zero of multiplicity3 ​, and 2 as a zero of multiplicity 1.

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  1. Answer:

    The polynomial is [tex]p(x) = ax^3(x+2)^3(x-2)[/tex], in which a is the leading coefficient.

    Step-by-step explanation:

    Zeros of a function:

    Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x – x_{1})*(x – x_{2})*…*(x-x_n)[/tex], in which a is the leading coefficient.

    -2 as a zero of multiplicity​ 3

    This means that:

    [tex]p(x) = (x-(-2))^3 = (x+2)^3[/tex]

    0 as a zero of multiplicity 3  

    Then also:

    [tex]p(x) = (x+2)^3(x-0)^3 = x^3(x+2)^3[/tex]

    2 as a zero of multiplicity 1.

    Then:

    [tex]p(x) = x^3(x+2)^3(x-2)[/tex]

    Adding the leading coefficient:

    [tex]p(x) = ax^3(x+2)^3(x-2)[/tex]

    The polynomial is [tex]p(x) = ax^3(x+2)^3(x-2)[/tex], in which a is the leading coefficient.

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