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The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation. Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve.
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.
For instance, the polynomial x 2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a 1 = −2 and a 2 = −1 of the above system gives the trinomial factorization: x 2 + 3x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1). The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
The polynomial () (+) is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes + +, with highest exponent 3.. The polynomial (+ +) + (+ + +) is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving + + + +, with highest exponent 5.
Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form:
Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient c n , {\displaystyle c_{n},} one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form + + + + =, where a ≠ 0. [1] The derivative of a quartic function is a cubic function.
Completing the cube is a similar technique that allows to transform a cubic polynomial into a cubic polynomial without term of degree two. More precisely, if + + + is a polynomial in x such that , its two first terms are the two first terms of the expanded form of