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The discriminant Δ of the cubic is the square of = () (), where a is the leading coefficient of the cubic, and r 1, r 2 and r 3 are the three roots of the cubic. As Δ {\displaystyle {\sqrt {\Delta }}} changes of sign if two roots are exchanged, Δ {\displaystyle {\sqrt {\Delta }}} is fixed by the Galois group only if the Galois group is A 3 .
The graph of any cubic function is similar to such a curve. The graph of a cubic function is a cubic curve, though many cubic curves are not graphs of functions. Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of ...
The coefficient of is ... The graphs of these polynomials ... [−1, 1] is simply given by the leading expansion coefficient ...
The graph of the zero polynomial, f(x) = 0, is the x-axis. ... (its leading coefficient) times a product of such polynomial factors of degree 1; ...
The 7th sum is indistinguishable from the original function at the resolution of the graph. In the appropriate Sobolev space , the set of Chebyshev polynomials form an orthonormal basis , so that a function in the same space can, on −1 ≤ x ≤ 1 , be expressed via the expansion: [ 16 ] f ( x ) = ∑ n = 0 ∞ a n T n ( x ) . {\displaystyle ...
The first (greatest) term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called the leading term, leading monomial and leading coefficient and denoted, in this article, lt(p), lm(p) and lc(p). Most polynomial operations related to Gröbner bases involve the leading terms.
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer.
If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients; If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q; If the degree of p is less than the degree of q, the limit is 0.