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Animation depicting the process of completing the square. (Details, animated GIF version)In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1]
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 .
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
Pascal's pyramid's first five layers. Each face (orange grid) is Pascal's triangle. Arrows show derivation of two example terms. In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. [1]
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as ...
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.
The third power of the trinomial a + b + c is given by (+ +) = + + + + + + + + +. This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
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