Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree: 4; zeros: 2i and -3i

Question

Form a polynomial f(x) with real coefficients having the given degree and zeros.  Degree: 4; zeros: 2i and -3i

kinggeorge · Answer

In general, if you have a polynomial with real coefficients, and some complex root $\alpha i$ where $\alpha$ is a real number, the you also have the complex root $-\alpha i$. Using this, can you tell me what all the roots of your polynomial will be?

anonymous · Answer

2i, -2i, -3i, and 3i?

kinggeorge · Answer

Bingo. So that means your factored polynomial will be $$(x-2i)(x+2i)(x-3i)(x+3i).$$Now you just have to expand it out.

anonymous · Answer

i got this answer: x^3 -3ix^2-2ix^2+6ix^2-4xi^2+12i^2.

kinggeorge · Answer

Hmmm. I've definitely got something different. I'll walk you through the first few steps I did.
$$(x-2i)(x+2i)=x^2-2ix+2ix-(2i)^2=x^2-4(-1)=x^2+4$$Note that $i^2=-1$ by definition.

Similarly, $$(x-3i)(x+3i)=x^2-3ix+3ix-(3i)^2=x^2-9(-1)=x^2+9$$Using this, can you find $$(x^2+4)(x^2+9)$$on your own?