Form a polynomial function with a degree of 4, zeros 2-4i; 4 multiplicity 2. Please help!

Question

Form a polynomial function with a degree of 4, zeros 2-4i; 4 multiplicity 2.  Please help!

anonymous · Answer

i will show u the steps then u conclude

anonymous · Answer

Perfect!!! Thank you very much!!!!!!!

anonymous · Answer

step one: you can get a zero from multiplying the complex number by its conjugant

anonymous · Answer

so in your case we have:

anonymous · Answer

$$(2-4i), (2+4i)$$

anonymous · Answer

sorry am on the phone i will continue the whole problem after am done

anonymous · Answer

No problem!  I had (x-(2-4i)(x-(2+i)(x+4)^2

anonymous · Answer

yes that is correct apart from a missing 4 in ur cojugent

anonymous · Answer

so u are having a problem multiplying them

anonymous · Answer

I get to this point and I'm c lost! I don't know what to do next!

anonymous · Answer

well u have to multiply the whole thing to get the polynomial

anonymous · Answer

the best way to do that is to simplify the product let me show u

anonymous · Answer

i am writing a simple method for u to always be able to solve such a problem quickly

anonymous · Answer

$$(x+iy)(x-iy)=x^2-(iy)^2$$

anonymous · Answer

r u there?

anonymous · Answer

I'm here sorry :-)

anonymous · Answer

$$[x-(2-4i)][x-2+(2-4i)][(x+4)^2]$$

anonymous · Answer

now let's solve the complex values first

anonymous · Answer

$$=[(x-2)+4i)][(x-2)-4i)]$$

anonymous · Answer

can u see i rearranged it in terms of real and imaginary parts ok

anonymous · Answer

$$=(x-2)^2-(4i)^2$$

anonymous · Answer

Yes I follow that

anonymous · Answer

$$=x^2-4x+4+16=x^2-4x+20$$

anonymous · Answer

Oh yeah that formula you gave earlier makes sense now

anonymous · Answer

$$=[x^2-4x+20][(x+4)^2]$$

anonymous · Answer

now lets expand

anonymous · Answer

so u can see we are getting a 4th order polynomial

anonymous · Answer

$$=[x^2-4x+20][x^2+8x+16]$$

anonymous · Answer

the final answer hence is:

anonymous · Answer

$$=x^4+4x^3+4x^2+96x+320$$

anonymous · Answer

i did that on piece of paper

anonymous · Answer

I was still calculating mine, but  I got the same answer!

anonymous · Answer

Thank you so very much for taking the time to actually walk me through step by step!!!!!!!

anonymous · Answer

good all the best

anonymous · Answer

don't forget my medal hehehe

anonymous · Answer

How do I do it

anonymous · Answer

how do i know hehehe, i really don't know

anonymous · Answer

i am just kidding anyways

anonymous · Answer

Figured it out :-) there you go, you have a medal !!! Thanks again!!!!!