Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those lis

Question

Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed
2(multiplicity 2), 3 + i (multiplicity 1)

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    2021-10-13T20:54:08+00:00

    Answer-

    The polynomial function is,

    y=x^4-10x^3+38x^2-64x+40

    Solution-

    The zeros of the polynomial are 2 and (3+i). Root 2 has multiplicity of 2 and (3+i) has multiplicity of 1

    The general form of the equation will be,

    \Rightarrow y=(x-(2))^2(x-(3+i))(x-(3-i))   ( ∵ (3-i) is the conjugate of (3+i) )

    \Rightarrow y=(x-2)^2(x-3-i)(x-3+i)

    \Rightarrow y=(x^2-4x+4)((x-3)-i)((x-3)+i)

    \Rightarrow y=(x^2-4x+4)((x-3)^2-i^2)

    \Rightarrow y=(x^2-4x+4)((x^2-6x+9)+1)

    \Rightarrow y=(x^2-4x+4)(x^2-6x+10)

    \Rightarrow y=x^2x^2-6x^2x+10x^2-4x^2x+4\cdot \:6xx-4\cdot \:10x+4x^2-4\cdot \:6x+4\cdot \:10

    \Rightarrow y=x^4-10x^3+14x^2+24x^2-40x-24x+40

    \Rightarrow y=x^4-10x^3+38x^2-64x+40

    Therefore, this is the required polynomial function.

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