## steve816 one year ago Please help me! Write a polynomial of degree 4 with zeros 3i, and -2 (multiplicity 2). Expand it out.

1. zepdrix

If a polynomial has a zero at -2 that means one of our solutions is:$\large\rm x=-2$Adding 2 to each side gives us,$\large\rm (x+2)=0$Our 4th degree polynomial will contain this as one of it's factors. Since this zero has a multiplicity of 2, it will show up twice in our polynomial.$\large\rm (x+2)(x+2)=0$Ok we're half way there!

2. zepdrix

3i is a zero, same idea,$\large\rm x=3i$$\large\rm (x-3i)=0$So we can add this factor to our polynomial that we're constructing,$\large\rm (x+2)(x+2)(x-3i)=0$

3. zepdrix

Recall that complex zeros always come in conjugate pairs. So if 3i is a zero of this polynomial, then -3i is also a zero.$\large\rm x=-3i$$\large\rm (x+3i)=0$Adding the last factor to our polynomial gives us$\large\rm (x+2)(x+2)(x-3i)(x+3i)=0$

4. zepdrix

And then you have to expand out all of that madness :p

5. steve816

Wow, great explanation zepdrix. I appreciate your work :)