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A direct corollary of the fundamental theorem of algebra is that any non-constant polynomial with real coefficients can be factored into a product of linear factors and irreducible quadratic factors, all with real coefficients. The linear factors correspond to the real roots, while the irreducible quadratic factors correspond to pairs of complex conjugate roots [latex]a pm bi[/latex].
This corollary bridges the gap between the abstract world of complex roots and the practical applications involving real numbers. The fundamental theorem guarantees that a real polynomial [latex]p(x)[/latex] of degree [latex]n[/latex] has [latex]n[/latex] complex roots. A key additional property is that if a polynomial has only real coefficients, its non-real roots must come in conjugate pairs. That is, if [latex]z = a + bi[/latex] is a root, then its conjugate [latex]bar{z} = a – bi[/latex] must also be a root. This can be shown by observing that [latex]p(bar{z}) = overline{p(z)}[/latex] for a real polynomial; if [latex]p(z)=0[/latex], then [latex]overline{p(z)}=0[/latex], so [latex]p(bar{z})=0[/latex].
Each pair of conjugate roots [latex](z, bar{z})[/latex] can be combined to form a real quadratic factor: [latex](x – z)(x – bar{z}) = (x – (a+bi))(x – (a-bi)) = x^2 – 2ax + (a^2+b^2)[/latex]. This quadratic has real coefficients and is irreducible over the real numbers because its discriminant is negative ([latex](-2a)^2 – 4(a^2+b^2) = -4b^2 < 0[/latex] for [latex]b neq 0[/latex]). By grouping all real roots into linear factors [latex](x-r)[/latex] and all conjugate pairs into irreducible quadratic factors, any real polynomial can be fully factored using only real coefficients. This result is immensely practical, especially in integral calculus for the decomposition of rational functions.
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実数多項式の因数分解
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