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Polynomial ring
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===Derivation=== {{main|Formal derivative|Derivation (differential algebra)}} The [[formal derivative|(formal) derivative]] of the polynomial :<math>a_0+a_1X+a_2X^2+\cdots+a_nX^n</math> is the polynomial :<math>a_1+2a_2X+\cdots+na_nX^{n-1}.</math> In the case of polynomials with [[real number|real]] or [[complex number|complex]] coefficients, this is the standard [[derivative]]. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of [[limit (mathematics)|limit]] is defined. The derivative makes the polynomial ring a [[differential algebra]]. The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers. ====Square-free factorization==== {{main|Square-free polynomial}} A polynomial with coefficients in a field or integral domain is ''square-free'' if it does not have a [[multiple root]] in the [[algebraically closed field]] containing its coefficients. In particular, a polynomial of degree {{mvar|n}} with real or complex coefficients is square-free if it has {{mvar|n}} distinct complex roots. Equivalently, a polynomial over a field is square-free if and only if the [[Polynomial greatest common divisor|greatest common divisor]] of the polynomial and its derivative is {{math|1}}. A ''square-free factorization'' of a polynomial is an expression for that polynomial as a product of powers of [[pairwise relatively prime]] square-free factors. Over the real numbers (or any other field of [[characteristic 0]]), such a factorization can be computed efficiently by [[Yun's algorithm]]. Less efficient algorithms are known for [[Factorization_of_polynomials_over_finite_fields#Square-free_factorization|square-free factorization of polynomials over finite fields]]. ====Lagrange interpolation==== {{main|Lagrange polynomial}} Given a finite set of ordered pairs <math>(x_j, y_j)</math> with entries in a field and distinct values <math>x_j</math>, among the polynomials <math>f(x)</math> that interpolate these points (so that <math>f(x_j) = y_j</math> for all <math>j</math>), there is a unique polynomial of smallest degree. This is the ''Lagrange interpolation polynomial'' <math>L(x)</math>. If there are <math>k</math> ordered pairs, the degree of <math>L(x)</math> is at most <math>k - 1</math>. The polynomial <math>L(x)</math> can be computed explicitly in terms of the input data <math>(x_j, y_j)</math>. ====Polynomial decomposition==== {{main|Polynomial decomposition}} A ''decomposition'' of a polynomial is a way of expressing it as a [[function composition|composition]] of other polynomials of degree larger than 1. A polynomial that cannot be decomposed is ''indecomposable''. [[Ritt's polynomial decomposition theorem]] asserts that if <math>f = g_1 \circ g_2 \circ \cdots \circ g_m = h_1 \circ h_2 \circ \cdots\circ h_n</math> are two different decompositions of the polynomial <math>f</math>, then <math>m = n</math> and the degrees of the indecomposables in one decomposition are the same as the degrees of the indecomposables in the other decomposition (though not necessarily in the same order).
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