[Math] Non- zero polynomial can be written as product of polynomials of degree at most two

Question

Prove that every non-zero polynomial $\in \mathbb{R}[x]$ can be written as product of polynomials in $\mathbb{R}[x]$ of degree at most two.

Is there elementary proof, at high school level ?

Please, suggest me on how to prove or lead me to the link.Thank you.

(Fundamental theorem of Algebra is quite hard for me.)

Answer 1

We need to assume that if $n$ is a degree of our polynomial $f$ then $f$ has $n$ roots from $\mathbb C$.

Try to prove the following statements.

• If $z\in\mathbb C$ is a root of our polynomial then $\bar{z}$ is a root of $f$.

• If previous $z=a+bi$, where $a$ and $b$ are reals, then $f$ divided by $x^2-2ax+a^2+b^2$.

• If $c\in\mathbb R$ is a root of $f$ thus, $f$ divided by $x-c$

  and by induction we are done!