# Factor polynomial into linear factors with complex coefficients.

Question: A polynomial is given. $(a)$ Factor it into linear and irreducible quadratic factors with real coefficients. $(b)$ Factor it completely into linear factors with complex coefficients.

$x^3 - 5x^2 + 4x - 20$

I factored it using the Rational Zeros Theorem and got the following expression: $(x-5)(x^2 + 4)$. Now, I think this is the answer for the 1st question, but how do I get complex coefficients? I can think of complex factors like $(x-5)(x-2i)(x+2i)$, but complex coefficients?

• You are asked to factor it completely, so the correct expression is $(x-5)(x-2i)(x+2i)$ which has indeed got complex coefficients. – almagest May 29 '16 at 8:34
• Aren't those factors and not coefficients? \ – MathEnthusiast May 29 '16 at 8:42
• $2i$ is a coefficient of the factor $x+2i$. – almagest May 29 '16 at 8:43

Decompose $P(x)=x^3-5x^2+4x-20$ over the complex field.