Factor polynomial into linear factors with complex coefficients.

Question

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?

Answer 1

The expression 'complex coefficients' was not the best way of describing what they wanted. The better way to phrase the question would have been

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

In which case your final result is what they want.