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Print/export Download as PDF ... move to sidebar hide. A perfect square is an element of algebraic structure that is equal to the square ... Perfect square trinomials
The effect has been to fold up the u 4 term into a perfect square: (u 2 + a) 2. The second term, au 2 did not disappear, but its sign has changed and it has been moved to the right side. The next step is to insert a variable y into the perfect square on the left side of equation , and a corresponding 2y into the coefficient of u 2 in the
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 ...
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To factorize the initial polynomial, it suffices to factorize each square-free factor. Square-free factorization is therefore the first step in most polynomial factorization algorithms. Yun's algorithm extends this to the multivariate case by considering a multivariate polynomial as a univariate polynomial over a polynomial ring.
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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.
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