Write a polynomial of sixth degree from known zeros

Question

Write a sixth degree polynomial with real coefficients.

Known zeros:

$$ 3+2i , -2, ..and ..0(third..degree) $$

we know that at $$p(1)=8$$

I tried the following: $$p(x)=(x-3-2i)(x+2)(x-0)^3(x-x_3)$$

Then I calculated for $x_3$ from $p(1)=8$.

Is this procedure correct?

The solution for this problem is: $$\frac13(x^6-4x^5+x^4+26x^3)$$

Answer 1

We have $$ p(x)=a(x-3-2i)(x-3+2i)(x+2)x^3 $$ The condition $p(1)=8$ is equivalent to $$ 3a-1=0. $$

Answer 2

Since $3+2i$ is a root, $3-2i$ is also a root, as polynomials with real coefficients can only have complex conjugates if one solution is a complex number.

$[x-(3+2i)][x-(3-2i)]=[(x-3)-2i][(x-3)+2i] = (x-3)^2+4=x^2-6x+13$

I think you can figure out the rest with your other method.