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  2. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    A cubic formula for the roots of the general cubic equation (with a ≠ 0) + + + = can be ... There is an interesting geometrical relationship among all these roots.

  3. Vieta's formulas - Wikipedia

    en.wikipedia.org/wiki/Vieta's_formulas

    Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when a n {\displaystyle a_{n}} is not a zero-divisor and P ( x ) {\displaystyle P(x)} factors as a n ( x − r 1 ) ( x − r 2 ) …

  4. Cube root - Wikipedia

    en.wikipedia.org/wiki/Cube_root

    Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number).

  5. Casus irreducibilis - Wikipedia

    en.wikipedia.org/wiki/Casus_irreducibilis

    Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced to the computation of square and cube roots. Cardano's formula for solution in ...

  6. Cubic function - Wikipedia

    en.wikipedia.org/wiki/Cubic_function

    The derivative of a cubic function is a quadratic function. A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point.

  7. Quartic function - Wikipedia

    en.wikipedia.org/wiki/Quartic_function

    For a general formula that is always true, one thus needs to choose a root of the cubic equation such that m ≠ 0. This is always possible except for the depressed equation y 4 = 0. Now, if m is a root of the cubic equation such that m ≠ 0, equation becomes

  8. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    In particular, the real roots are mostly located near ±1, and, moreover, their expected number is, for a large degree, less than the natural logarithm of the degree. If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula [21] [22]

  9. Cubic equations of state - Wikipedia

    en.wikipedia.org/wiki/Cubic_equations_of_state

    This equation may have up to three roots. The maximal root of the cubic equation generally corresponds to a vapor state, while the minimal root is for a liquid state. This should be kept in mind when using cubic equations in calculations, e.g., of vapor-liquid equilibrium.