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14 September, 02:03

according to the fundamental theorem of algebra how many roots exits for the polynomial function. f (x) = 8x^7-x^5+x^3+6

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  1. 14 September, 02:32
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    The Fundamental Theorem of Algebra (FTOA) tells us that any non-zero polynomial in one variable with complex (possibly real) coefficients has a complex zero.

    A straightforward corollary, often stated as part of the FTOA is that a polynomial in a single variable of degree

    n

    >

    0

    with complex (possibly real) coefficients has exactly

    n

    complex (possibly real) zeros, counting multiplicity.

    To see that the corollary follows, note that if

    f

    (

    x

    )

    is a polynomial of degree

    n

    >

    0

    and

    f

    (

    a

    )

    =

    0

    , then

    (

    x

    -

    a

    )

    is a factor of

    f

    (

    x

    )

    and

    f

    (

    x

    )

    x

    -

    a

    is a polynomial of degree

    n

    -

    1

    . So repeatedly applying the FTOA, we find that

    f

    (

    x

    )

    has exactly

    n

    complex zeros counting multiplicity.

    Discriminants

    If you want to know how many real roots a polynomial with real coefficients has, then you might like to look at the discriminant - especially if the polynomial is a quadratic or cubic. Ths discriminant gives less information for polynomials of higher degree.

    The discriminant of a quadratic

    a

    x

    2

    +

    b

    x

    +

    c

    is given by the formula:

    Δ

    =

    b

    2

    -

    4

    a

    c

    Then:

    Δ

    >

    0

    indicates that the quadratic has two distinct real zeros.

    Δ

    =

    0

    indicates that the quadratic has one real zero of multiplicity two (i. e. a repeated zero).

    Δ

    <

    0

    indicates that the quadratic has no real zeros. It has a complex conjugate pair of non-real zeros.

    The discriminant of a cubic

    a

    x

    3

    +

    b

    x

    2

    +

    c

    x

    +

    d

    is given by the formula:

    Δ

    =

    b

    2

    c

    2

    -

    4

    a

    c

    3

    -

    4

    b

    3

    d

    -

    27

    a

    2

    d

    2

    +

    18

    a

    b

    c

    d

    Then:

    Δ

    >

    0

    indicates that the cubic has three distinct real zeros.

    Δ

    =

    0

    indicates that the cubic has either one real zero of multiplicity

    3

    or one real zero of multiplicity

    2

    and another real zero.

    Δ

    <

    0

    indicates that the cubic has one real zero and a complex conjugate pair of non-real zeros.
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