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In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix. 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.
Therefore, if a second degree integer polynomial factor exists, it must take one of the values p(0) = 1, 2, −1, or −2. and likewise for p(1). There are eight factorizations of 6 (four each for 1×6 and 2×3), making a total of 4×4×8 = 128 possible triples (p(0), p(1), p(−1)), of which half can be discarded as the negatives of the other ...
Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n 2) operations in F q using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in F q using "fast" arithmetic.
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the ...
Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors of degree no higher than 2: just 1st-degree and quadratic factors.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
The Cantor–Zassenhaus algorithm takes as input a square-free polynomial (i.e. one with no repeated factors) of degree n with coefficients in a finite field whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, () / ((), ′ ()) is a squarefree ...