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Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when a n {\displaystyle a_{n}} is not a zero-divisor and P ( x ) {\displaystyle P(x)} factors as a n ( x − r 1 ) ( x − r 2 ) …
In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first n terms in the limit gives an expression for π that is accurate to approximately 0.6n digits. [4] [15] This convergence rate compares very favorably with the Wallis product, a later infinite product formula for π.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
As (+) = and (+) + =, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator , by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation ).
This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a. It also follows from dividing the quadratic equation by x 2 {\displaystyle x^{2}} giving c x − 2 + b x − 1 + a = 0 , {\displaystyle cx^{-2}+bx^{-1}+a=0,} solving this for x − 1 , {\displaystyle x^{-1},} and then ...
There is a formula [7] for calculating the Möbius function without directly knowing the factorization of its argument: = (,) =, i.e. () is the sum of the primitive -th roots of unity. (However, the computational complexity of this definition is at least the same as that of the Euler product definition.)
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
In mathematics, a sum of radicals is defined as a finite linear combination of n th roots: =, where , are natural numbers and , are real numbers.. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time.