$x^3-3x^2+4x-2$ cannot be factored over $\mathbb R$ I'm new to the site, and I need a bit of help from you.
How can I prove that the polynomial:  $f(x)=x^3-3x^2+4x-2$
cannot be factored as a product of polynomials of degree 1 with real coefficients?
Thanks.

Observe that$$\begin{align*}
f(x)=x^3-3x^2+4x-2&=(x^3-3x^2+3x-1)+(x-1)\\\\
&=(x-1)^3+(x-1)\\\\
&=(x-1)((x-1)^2+1)\\\\
&=(x-1)(x^2-2x+2)
\end{align*}$$
By the quadratic formula, the roots of $x^2-2x+2$ are complex, so $f$ cannot be factored any further into polynomials with real coefficients.
