The Fundamental Theorem of Algebra

Definition A complex polynomial is an expression

where $n$ is a nonnegative integer and the coefficients $a_0, \ldots, a_n$ are complex numbers. If $a_n \ne 0$, the degree is $n$ and $a_n$ is the leading coefficient. The polynomial is nonconstant when its degree is at least $1$. A zero of $P$ is a complex number $c$ with $P(c) = 0$. Real numbers are complex numbers, so every polynomial with real coefficients is also a complex polynomial.

Definition The Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficients has at least one zero in $\mathbb{C}$. The theorem is about existence. It does not give a formula for the zero, and for general polynomials of degree $5$ and higher no formula built from the coefficients with arithmetic and roots exists. Its proofs use ideas from beyond algebra, such as continuity.

Definition Every polynomial $P$ of degree $n \ge 1$ with complex coefficients can be written as

where $a_n$ is the leading coefficient and $c_1, \ldots, c_n$ are complex numbers, not necessarily different. The list $c_1, \ldots, c_n$ is unique apart from its order, and it consists exactly of the zeros of $P$. The proof repeats one step. The theorem gives a zero $c_1$, the Factor Theorem gives $P(z) = (z - c_1)Q(z)$ with $Q$ of degree $n - 1$, and the same argument applies to $Q$ until a constant is left.

Definition Let $P$ be a nonzero polynomial and $c$ a complex number. The multiplicity of $c$ is the largest integer $m \ge 0$ such that $(z - c)^m$ divides $P(z)$. The number $c$ is a zero exactly when $m \ge 1$. For a zero this is the multiplicity from the lesson on zeros and multiplicity, the exponent $m$ with $P(z) = (z - c)^m Q(z)$ and $Q(c) \ne 0$. In the complex factorization, the multiplicity of $c$ is the number of factors equal to $z - c$. It follows that a polynomial of degree $n \ge 1$ has exactly $n$ complex zeros when each zero is counted as many times as its multiplicity, and therefore at most $n$ different zeros.

Definition Let $P$ have real coefficients. If a complex number $c$ is a zero of $P$, then its conjugate $\overline{c}$ is also a zero, with the same multiplicity. The reason for the first part is that conjugation respects sums and products and leaves real coefficients unchanged, so

So the nonreal zeros of a real polynomial come in conjugate pairs. As a consequence, a polynomial with real coefficients and odd degree has at least one real zero. The rule needs real coefficients. The polynomial $z - i$ has the zero $i$ but not $-i$.

Definition A quadratic $ax^2 + bx + c$ with real coefficients and $a \ne 0$ is irreducible over the reals when it is not a product of two linear factors with real coefficients. This happens exactly when it has no real zeros, that is when $b^2 - 4ac < 0$. For a complex number $w$, multiplying the linear factors of the pair $w$ and $\overline{w}$ always gives a quadratic with real coefficients,

and it is irreducible when $w$ is not real. Grouping each conjugate pair this way shows that every polynomial with real coefficients and degree at least $1$ is its leading coefficient times a product of real linear factors $x - r$ and real irreducible quadratic factors $x^2 + px + q$ with $p^2 - 4q < 0$.