# All Polynomials With Real Coefficients Can Be Factored Into Linear And Irreducible Quadratic Terms All polynomials with real coefficients can be factored into linear terms with real coefficients, and irreducible quadratic terms with real coefficients. An irreducible polynomial here means that the polynomial cannot be further factored into linear terms with real coefficients. 1. As per [[The Fundamental Theorem of Algebra]], all $n$ degree polynomials can be factored into $n$ linear terms of the form $x-c$ where $c$ is the polynomial's root. 2. If $c$ is a complex number, and the polynomial has real coefficients, $x- \overline c$ is also a factor.[^1] 3. If we multiply $(x-c)(x-\overline c)$ we get an irreducible[^2] quadratic polynomial with real coefficients (see below for proof). Consequently, all linear terms of a polynomial with real coefficients either correspond to real roots (in which case they have real coefficients by definition), or complex roots, in which case they come in pairs that, when multiplied, produce irreducible quadratic terms with real coefficients. [^1]: [[A Complex Conjugate of a Root of a Polynomial with Real Coefficients Is Also a Root]] [^2]: Per definition of 'irreducible', see above ## Multiplying $(x-c)(x-\overline c)$ Produces a Polynomial With Real Coefficients $ \begin{align} (x-c)(x-\overline c) &= x^2 - x\overline c -xc + c \overline c \\ &=x^2-x(c + \overline c) + c \overline c \end{align} $ $(c + \overline c)$ is a real number by definition of a conjugate, and $c \overline c$ is equal to the square of the magnitude, which is also real per definition.[^3] [^3]: See [[Complex Numbers#Identities]] for proofs.