The Fundamental Theorem of Algebra states that any real-valued polynomial `p(x)` of degree `n > 2` can be divided by some quadratic polynomial with real coefficients. To prove this, we can use a Sturm-like chain that contains the coefficients of the polynomial, `Rp(x)(a, b)` and `Sp(x)(a, b)`, as consecutive terms. By taking a sufficiently large negative value of `b`, we can show that the roots of both `Rp(x)(a, b)` and `Sp(x)(a, b)` in the variable `a` are real-valued and alternating each other (interlacing property). This interlacing property can then be used to show that the locus of the roots of `Rp(x)(a, b)` and `Sp(x)(a, b)` must intersect for some real-valued `a` and `b < 0`. Therefore, the Fundamental Theorem of Algebra is proven.
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