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In mathematics, a Thue equation is a Diophantine equation of the form (,) =,where is an irreducible bivariate form of degree at least 3 over the rational numbers, and is a nonzero rational number.
For instance, the polynomial x 2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a 1 = −2 and a 2 = −1 of the above system gives the trinomial factorization: x 2 + 3x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1). The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has 77 = 7 · 11, and thus n = 2 · 3 2 · 7 · 11. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.
The computation of Q and its roots can be used to solve P(x) = 0. However these p-th roots may not be computed independently (this would provide p p–1 roots instead of p). Thus a correct solution needs to express all these p-roots in term of one of them. Galois theory shows that this is always theoretically possible, even if the resulting ...
For each solution (c 0, s 0) of this system, there is a unique solution x of the equation such that 0 ≤ x < 2 π. In the case of this simple example, it may be unclear whether the system is, or not, easier to solve than the equation.