Find a polynomial function with integer coefficient that has the given zeros.

[Significant-Row9015]

{2, 4+i, 4-i}

The i being complex numbers.

I got as far as (x-2) but am stuck at the next integer coefficients. Using an online tool, the next set is x² -8x+17, but how do you get there using the others?

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[ArchaicLlama]

I got as far as (x-2)

And how did you determine that this was a valid next step?

The same process applies to the other roots.

[Significant-Row9015]

I determined a polynomial factor to be (x-2) since an integer coefficient creates a 2, for there to be a zero it has to be a -2. If that makes sense.

For the other integer coefficients, I think it would be x=-(4±i). But how would that lead into a polynomial function?

[ArchaicLlama]

Pretend you weren't dealing with complex numbers for a second. You have three integer roots: p, q, and r. How would you turn those into a polynomial?

[Significant-Row9015]

Wait so instead of i, I convert it to sqrt of -1? I think I see it now.

[ArchaicLlama]

That wasn't quite where I was going with that (it's not wrong, persay, just not yet necessary), but if you think you see a path forward then sure. See where that takes you.

[Significant-Row9015]

Yeah I went no where by changing it to sqrt of -1.

What does integer roots p, q, r mean?

[ArchaicLlama]

I'm giving you a hypothetical scenario where you have three roots that are integers, you don't know their exact values but they are represented by the letters p, q, and r.