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A perfect square is an element of algebraic structure that is equal to the square of another element. ... Perfect square trinomials, a method of factoring polynomials
Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial – the number of terms is clearly a triangular number. In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
These factorizations work not only over the complex numbers, but also over any field, where either –1, 2 or –2 is a square. In a finite field , the product of two non-squares is a square; this implies that the polynomial x 4 + 1 , {\displaystyle x^{4}+1,} which is irreducible over the integers, is reducible modulo every prime 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.
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
To find integer solutions to + =, find positive integers r, s, and t such that = is a perfect square. Then: = +, = +, = + +. From this we see that r is any even integer and that s and t are factors of r 2 /2. All Pythagorean triples may be found by this method.
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The naive approach to finding a congruence of squares is to pick a random number, square it, divide by n and hope the least non-negative remainder is a perfect square. For example, 80 2 ≡ 441 = 21 2 ( mod 5959 ) {\displaystyle 80^{2}\equiv 441=21^{2}{\pmod {5959}}} .