Fundamental Theorem of Algebra

Key Questions

  • How do you know how many roots are in a polynomial?

    Answer:

    See explanation...

    Explanation:

    Fundamental Theorem of Algebra

    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.

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    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 #ax^2+bx+c# is given by the formula:

    #Delta = b^2-4ac#

    Then:

    #Delta > 0# indicates that the quadratic has two distinct real zeros.

    #Delta = 0# indicates that the quadratic has one real zero of multiplicity two (i.e. a repeated zero).

    #Delta < 0# indicates that the quadratic has no real zeros. It has a complex conjugate pair of non-real zeros.

    The discriminant of a cubic #ax^3+bx^2+cx+d# is given by the formula:

    #Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#

    Then:

    #Delta > 0# indicates that the cubic has three distinct real zeros.

    #Delta = 0# indicates that the cubic has either one real zero of multiplicity #3# or one real zero of multiplicity #2# and another real zero.

    #Delta < 0# indicates that the cubic has one real zero and a complex conjugate pair of non-real zeros.

    George C. · 6 · Sep 9 2017

Questions

Complex Zeros