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The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra , which does not provide any tool for computing exactly the solutions, although Newton's method allows ...
A rational quadratic form in five or more variables represents zero over the field ℚ p of p-adic numbers for all p. Meyer's theorem is the best possible with respect to the number of variables: there are indefinite rational quadratic forms Q in four variables which do not represent zero. One family of examples is given by Q(x 1,x 2,x 3,x 4 ...
In other words, it was necessary to prove only that the equation a n + b n = c n has no positive integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The general equation a ...
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.
If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.