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By the fundamental theorem of algebra, if the monic polynomial equation x 2 + bx + c = 0 has complex coefficients, it must have two (not necessarily distinct) complex roots. Unfortunately, the discriminant b 2 − 4c is not as useful in this situation, because it may be a complex number. Still, a modified version of the general theorem can be ...
If x 3 is the remaining fraction after this step of the greedy expansion, it satisfies the equation P 2 (x 3 + 1 / 9 ) = 0, which can again be expanded as a polynomial equation with integer coefficients, P 3 (x 3) = 324x 2 3 + 720x 3 − 5 = 0. Continuing this approximation process eventually produces the greedy expansion for the golden ...
The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
[17] [18] For example, the fraction 1/(x 2 + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x. For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers.
The greedy algorithm for Egyptian fractions finds a solution in three or fewer terms whenever is not 1 or 17 mod 24, and the 17 mod 24 case is covered by the 2 mod 3 relation, so the only values of for which these two methods do not find expansions in three or fewer terms are those congruent to 1 mod 24.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2x 2 + 4x − 4 = 0. Although the display shows only five significant figures of accuracy, the retrieved value of xc is 0.732050807569, accurate to twelve significant figures. A quadratic function without real root: y = (x − 5) 2 + 9.