if alpha beta gamma are the zeros of the polynomial Y is square minus 6 y square minus y + 30 then the value of alpha beta gamma​

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if alpha beta gamma are the zeros of the polynomial Y is square minus 6 y square minus y + 30 then the value of alpha beta gamma​

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Genesis 4 days 2021-10-10T07:19:06+00:00 1 Answer 0 views 0

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    2021-10-10T07:21:02+00:00

    The values of \alpha,\beta,\gamma are -2,3,5

    Step-by-step explanation:

    Given

    \alpha,\beta,\gamma are zeroes of polynomial

    y^3-6y^2-y+30

    To find out:

    Values of \alpha,\beta,\gamma

    Solution:

    Let

    p(y)=y^3-6y^2-y+30

    Putting y = 1

    p(1)=1^3-6\times 1^2-1+30=1-6-1+30=24\ne 0

    Putting y = -1

    p(-1)=(-1)^3-6\times (-1)^2-(-1)+30=-1-6+1+30=24\ne 0

    Putting y = -2

    p(-2)=(-2)^3-6\times (-2)^2-(-2)+30=-8-24+2+30=32-32=0

    Since on putting y = -2, the value of polynomial p(y) is zero

    Therefore, y = -2 is a zero of the polynomial p(y)

    Therefore, y+2 is a factor of polynomial p(y)

    Thus, p(y) can be written as

    p(y)=(y+2)(y^2-8y+15)

    \implies p(y)=(y+2)(y^2-5y-3y+15)

    \implies p(y)=(y+2)[y(y-5)-3(y-5)]

    \implies p(y)=(y+2)(y-3)(y-5)

    Thus,

    (y+2), (y-3), (y-5) are factors of polynomial p(y)

    Or, y = -2, y = 3, y = 5 are zeroes of polynomial p(y)

    Therefore, the values of \alpha,\beta,\gamma are -2,3,5

    Hope this answer is helpful.

    Know More:

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