Factorize polynomial in Matlab I want to factor (break up) a polynomial $P(x)$ into first orders $(x + a_i)$ for real roots and second orders $(x^2 + b_ix + c_i)$ for complex roots. That is to say, $$P(x) = \prod (x + a_i) \prod (x^2 + b_ix + c_i)$$ Herein $a_i, b_i, c_i \in \mathbb{R}$. How can I achieve this in Matlab? Matlab has the `roots` function but this gives all the roots in first orders as such, they are also complex. However, for the complex roots I want them to appear into second order.

If you have a polynomial with real coefficients, the roots are real or occur in complex conjugate pairs.

So one way to do this would be to call the roots function, and first remove all the results which are real (these will be the linear factors). Then, take all the factors which have non-zero imaginary part, and throw out the ones which have strictly negative imaginary part. You'll be left with roots $r_1,r_2,...$. Then, the quadratic factors will be given by $(x-r_i)(x-\bar{r_i}) = x - 2 Re(r_i) x + |r_i|^2$.