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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?

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  • $\begingroup$ You are asked to factor it completely, so the correct expression is $(x-5)(x-2i)(x+2i)$ which has indeed got complex coefficients. $\endgroup$
    – almagest
    May 29, 2016 at 8:34
  • $\begingroup$ Aren't those factors and not coefficients? \ $\endgroup$ May 29, 2016 at 8:42
  • $\begingroup$ $2i$ is a coefficient of the factor $x+2i$. $\endgroup$
    – almagest
    May 29, 2016 at 8:43

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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.

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