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In either case the full quartic can then be divided by the factor (x − 1) or (x + 1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots. If a 1 = a 0 k , {\displaystyle \ a_{1}=a_{0}k\ ,} a 2 = 0 {\displaystyle \ a_{2}=0\ } and a 4 = a 3 k , {\displaystyle \ a_{4}=a_{3}k\ ,} then x = − k ...
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...
For example, let a denote a multiplicative generator of the group of units of F 4, the Galois field of order four (thus a and a + 1 are roots of x 2 + x + 1 over F 4. Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0.
A perfect pair from a 1-factorization is a pair of 1-factors whose union induces a Hamiltonian cycle. A perfect 1-factorization (P1F) of a graph is a 1-factorization having the property that every pair of 1-factors is a perfect pair. A perfect 1-factorization should not be confused with a perfect matching (also called a 1-factor).
Each r is a norm of a − r 1 b and hence that the product of the corresponding factors a − r 1 b is a square in Z[r 1], with a "square root" which can be determined (as a product of known factors in Z[r 1])—it will typically be represented as an irrational algebraic number.
where R 1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q 1 is m×n, Q 2 is m×(m − n), and Q 1 and Q 2 both have orthogonal columns. Golub & Van Loan (1996 , §5.2) call Q 1 R 1 the thin QR factorization of A ; Trefethen and Bau call this the reduced QR factorization . [ 1 ]
Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus.It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics.
To factorize the initial polynomial, it suffices to factorize each square-free factor. Square-free factorization is therefore the first step in most polynomial factorization algorithms. Yun's algorithm extends this to the multivariate case by considering a multivariate polynomial as a univariate polynomial over a polynomial ring.
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