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The coefficient attached to the highest degree of the variable in a polynomial of one variable is referred to as the leading coefficient; for example, in the example expressions above, the leading coefficients are 2 and a, respectively.
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, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
q is an integer factor of the leading coefficient a n. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.
The Chebyshev polynomials T n are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.
It may happen that this makes the coefficient 0. [12] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, [d] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. A real polynomial is a polynomial with real coefficients.
It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis).
The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients (Schenck 2003, pp. 41). All these definitions may be extended to finitely generated graded modules over S , with the only difference that a factor t m appears in the Hilbert series, where m is the minimal ...
With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of given below. This definition of the P n {\displaystyle P_{n}} 's is the simplest one.