Question

Please help me!

Write a polynomial of degree 4 with zeros 3i, and -2 (multiplicity 2). Expand it out.

Answer 1

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!

Answer 2

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,\[\large\rm (x+2)(x+2)(x-3i)=0\]

Answer 3

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\[\large\rm (x+2)(x+2)(x-3i)(x+3i)=0\]

Answer 4

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

Answer 5

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