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The Fundamental Theorem of Algebra

Statement of Theorem

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Two important results follow from this:

  • Every single-variable polynomial of degree $n$, $n \geq 1$, with complex coefficients has, counted with multiplicity, $n$ complex roots.
  • Every single-variable polynomial of degree $n$, $n \geq 1$, with real coefficients can be written as a product of polynomials with real coefficients whose degree is either 1 or 2. Hence the polynomial will have either no complex roots or an even number of complex roots, each complex root having its complex conjugate also as a root.

It follows from the last statement that if $p(x)=a_0+a_1x+\cdots+a_n x^n$ is a polynomial of degree $n$, $n \geq 1$, with real coefficients, then:

  • If $n$ is odd it must have an odd number of real roots (hence at least one real root).
  • If $n$ is even then it can have either no real roots or an even number of real roots.
Examples
  • A quadratic $q(x)=ax^2+bx+c$ where $a,\;b,\;c$ are real numbers can have $0$ or $2$ real roots. The number of real roots is given by the sign of the discriminant. A quadratic which has $0$ real roots cannot be further factorised and is called irreducible.
  • A cubic $p(x)=ax^3+bx^2+cx+d$ where $a,\;b,\;c,\;d$ are real numbers can have 1 or 3 real roots. So a cubic always has at least one real root. This means that $p(x)$ can always be rewritten $p(x)=(x-r)q(x)$, where $r \in \mathbb{R}$ and $q(x)$ is a quadratic which may have $0$ or $2$ real roots.

See Also

Whiteboard maths

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