# Write a polynomial function of least degree with integral coefficients that has the given zeros: `4` , `3i`

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### 1 Answer

Well, in order to have a polynomial with Real (let alone Integral) coefficients, you need to get rid of those complex numbers!

It turns out that the only way for you to have a polynomial without imaginary terms is to ensure that every complex root has its complex conjugate **as another root**.

How do I get the complex conjugate? Well, every complex number is of the form:

`a+bi`

where `a` is the "real part" and `b` is the "imarginary part." In order to construct the complex conjugate, we simply take the additive inverse **(negative) of the imaginary part**! For example, the complex conjugate to `a+bi` would be:

`a-bi`

How do these two help you get rid of imaginary numbers? Let me show you in general what happens to two roots that are complex conjugates:

`(x-(a+bi))(x-(a-bi)) = x^2 - ax + xbi -ax +a^2 - abi -xbi + abi -b^2i^2`

Now, let's simplify (remember, `i^2 = -1`) by putting all the real terms on one side, and all the imaginary terms on the other:

`x^2 - 2ax + a^2 + b^2 + i(-xb-ab+xb+ab)=x^2-2ax+a^2+b^2`

Notice that the imaginary side simply becomes zero! **This would not happen if the roots were not complex conjugates.**

So, how does all this apply to those two roots? Well, we are given one real root, and one imaginary root:

`x=4, x = 3i`

Well, remember how whenever we have an imaginary root, we need to take the negative of the imaginary part to make it a complex conjugate. Also, this complex conjugate is the only way to make the coefficients real. Therefore, **we have to add one more root** to this set of two in order to get real, integral coefficients. That additional root will be `-3i`.

So, our polynomial will become:

`(x-4)(x-3i)(x+3i) = (x-4)(x^2 + 3ix -3ix - 3^2i^2)`

Simplifying (again, make sure you account for `3^2i^2 = -9` :

`(x-4)(x^2+9) = x^3 -4x^2 +9x -36`

So there's our final polynomial. We have also found the minimum degree necessary to have integral coefficients: 3.

I hope that helps!

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