Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

中文翻译：

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使用多项式交错的严格实数代数基本定理

在不求助于复数或任何高级拓扑论证的情况下，我们证明了任何次数大于 2 的实数多项式总是具有实数二次多项式因子，这等价于代数基本定理。该证明使用交错的二元多项式，类似于高斯使用复数对代数基本定理的第一个证明，但在严格实数多项式的除法残差的不同上下文中。这表明基本实分析作为证明代数基本定理的最小平台是足够的。