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分解二次三項式的因式,一般說有四种方法:公式法、配方法、分組分解法和余式定理法。但对分組分解法作进一步研究又可得出观察法。用观察法分解某些(尤其是整系数的)二次三項式的因式时,有快而准的特点,因此在实际計算中我們經常用到它。但要采用观察法,必須在“B~2-4AC是某数的平方时,整系数二次式Ax~2+Bx+C一定是两整系数一次式之积。”这一命題正确的条件下方可。否則(有时可能出現分数),問題将变得复杂多了,不易“观察”。当然就談不上快而准了。所以,有証明这一命題正确的必要,本文的目的正是这样。引理Ⅰ.两奇数的平方差,必是8的倍数;奇数与偶数的平方差,必是奇数。
There are four methods for decomposing the factor of quadratic trinomial: formula method, matching method, group decomposition method, and remainder type theorem method. However, further research on the grouping decomposition method can also provide observations. Decomposition of some (especially integral coefficient) quadratic trinomial factors by observation method has a fast and accurate characteristic, so we often use it in the actual calculation. But to use the observation method, it must be “When B~2-4AC is the square of a certain number, the integral coefficient Ax~2+Bx+C must be the product of the two integral coefficients once.” This proposition is correct Conditions can be below. Otherwise (sometimes there may be points), the problem will become more complicated and difficult to “observe.” Of course, it’s not quick and accurate. Therefore, it is necessary to prove that this proposition is correct. The purpose of this paper is exactly this. Lemma I. The squared difference of two odd numbers must be a multiple of eight; the squared difference between odd and even numbers must be odd.