# How do you write a polynomial with zeros 6, 2 + 2i?

##### 1 Answer

#### Explanation:

If a polynomial has a zero at

Here, we see that the zero at

In polynomials with real coefficients, which I'm assuming this is, **complex roots always come in pairs**. The pairs of complex roots are always complex conjugates of one another. This means, that if

Constructing a polynomial with zeros of

#p(x)=(x-6)(x-(2+2i))(x-(2-2i))#

Multiplying the

#p(x)=(x-6)((x-2)-2i)((x-2)+2i)#

Now, the final two terms are in the form

#p(x)=(x-6)((x-2)^2-(2i)^2)#

#p(x)=(x-6)(x^2-4x+4-4i^2)#

Recall that

#p(x)=(x-6)(x^2-4x+4+4)#

#p(x)=(x-6)(x^2-4x+8)#

Distribute completely:

#p(x)=x^3-10x^2+32x-48#