Answer:
Standard form: ![x^5-14x^4+83x^3-256x^2+406x-260](https://tex.z-dn.net/?f=x%5E5-14x%5E4%2B83x%5E3-256x%5E2%2B406x-260)
Factored form with complex numbers: ![(x-2)(x-(3+i))(x-(3-i))(x-(3-2i))(x-(3+2i))](https://tex.z-dn.net/?f=%28x-2%29%28x-%283%2Bi%29%29%28x-%283-i%29%29%28x-%283-2i%29%29%28x-%283%2B2i%29%29)
Factored form without complex numbers: ![(x-2)(x^2-6x+10)(x^2-6x+13)](https://tex.z-dn.net/?f=%28x-2%29%28x%5E2-6x%2B10%29%28x%5E2-6x%2B13%29)
Step-by-step explanation:
If the polynomial has real coefficients and a+bi is a zero, then a-bi zero.
This means the following:
1) Since 3+i is a zero, then 3-i is a zero.
2) Since 3-2i is a zero, then 3+2i is a zero.
So we have the following zeros:
1) x=2
2) x=3+i
3) x=3-i
4) x=3-2i
5) x=3+2i
If x=c is a zero of the polynomial, then x-c is a factor of the polynomial.
This implies the following:
1) x=2 is a zero => x-2 is a factor
2) x=3+i is a zero => x-(3+i) is a factor
3) x=3-i is a zero => x-(3-i) is a factor
4) x=3-2i is a zero => x-(3-2i) is a factor
5) x=3+2i is a zero => x-(3+2i) is a factor
Let's put our factors together:
![(x-2)(x-(3+i))(x-(3-i))(x-(3-2i))(x-(3+2i))](https://tex.z-dn.net/?f=%28x-2%29%28x-%283%2Bi%29%29%28x-%283-i%29%29%28x-%283-2i%29%29%28x-%283%2B2i%29%29)
Let's find the standard form for this polynomial.
This will require us to multiply the above out and simplify by combining any like terms.
Before we begin this process, I would rather have a quick way to multiply the factors that contain the congregate pair zeros.
![(x-(m+ni))(x-(m-ni))](https://tex.z-dn.net/?f=%28x-%28m%2Bni%29%29%28x-%28m-ni%29%29)
I'm going to use foil.
First: ![x(x)=x^2](https://tex.z-dn.net/?f=x%28x%29%3Dx%5E2)
Outer: ![x(-(m-ni))=-(m-ni)x](https://tex.z-dn.net/?f=x%28-%28m-ni%29%29%3D-%28m-ni%29x)
Inner: ![-(m+ni)(x)=-(m+ni)x](https://tex.z-dn.net/?f=-%28m%2Bni%29%28x%29%3D-%28m%2Bni%29x)
Last: ![-(m+ni)(-(m-ni))=(m+ni)(m-ni)=m^2-n^2i^2=m^2-n^2(-1)=m^2+n^2](https://tex.z-dn.net/?f=-%28m%2Bni%29%28-%28m-ni%29%29%3D%28m%2Bni%29%28m-ni%29%3Dm%5E2-n%5E2i%5E2%3Dm%5E2-n%5E2%28-1%29%3Dm%5E2%2Bn%5E2)
(Note:
; When multiplying conjugates, you just have to multiply the first terms of each and the last terms of each.)
------------------------------------Let's combine these terms:
![x^2-(m-ni)x-(m+ni)x+m^2+n^2](https://tex.z-dn.net/?f=x%5E2-%28m-ni%29x-%28m%2Bni%29x%2Bm%5E2%2Bn%5E2)
Distribute:
![x^2-mx+nix-mx-nix+m^2+n^2](https://tex.z-dn.net/?f=x%5E2-mx%2Bnix-mx-nix%2Bm%5E2%2Bn%5E2)
Combine like terms:
![x^2-2mx+m^2+n^2](https://tex.z-dn.net/?f=x%5E2-2mx%2Bm%5E2%2Bn%5E2)
So the formula we will be using on the factors that contain conjugate pair zeros is:
.
![(x-(3+i))(x-(3-i))=x^2-2(3)x+3^2+1^2](https://tex.z-dn.net/?f=%28x-%283%2Bi%29%29%28x-%283-i%29%29%3Dx%5E2-2%283%29x%2B3%5E2%2B1%5E2)
![(x-(3+i))(x-(3-i))=x^2-6x+9+1](https://tex.z-dn.net/?f=%28x-%283%2Bi%29%29%28x-%283-i%29%29%3Dx%5E2-6x%2B9%2B1)
![(x-(3+i))(x-(3-i))=x^2-6x+10](https://tex.z-dn.net/?f=%28x-%283%2Bi%29%29%28x-%283-i%29%29%3Dx%5E2-6x%2B10)
![(x-(3+2i))(x-(3-2i))=x^2-2(3)x+3^2+2^2](https://tex.z-dn.net/?f=%28x-%283%2B2i%29%29%28x-%283-2i%29%29%3Dx%5E2-2%283%29x%2B3%5E2%2B2%5E2)
![(x-(3+2i))(x-(3-2i))=x^2-6x+9+4](https://tex.z-dn.net/?f=%28x-%283%2B2i%29%29%28x-%283-2i%29%29%3Dx%5E2-6x%2B9%2B4)
![(x-(3+2i))(x-(3-2i))=x^2-6x+13](https://tex.z-dn.net/?f=%28x-%283%2B2i%29%29%28x-%283-2i%29%29%3Dx%5E2-6x%2B13)
------------------------------------------------
So this is what we have now:
![(x-2)(x^2-6x+10)(x^2-6x+13)](https://tex.z-dn.net/?f=%28x-2%29%28x%5E2-6x%2B10%29%28x%5E2-6x%2B13%29)
I'm going to multiply the last two factors:
![(x^2-6x+10)(x^2-6x+13)](https://tex.z-dn.net/?f=%28x%5E2-6x%2B10%29%28x%5E2-6x%2B13%29)
What we are going to do is multiply the first term in the first ( ) to every term in the second ( ).
We are also going to do the same for the second term in the first ( ).
Then the third term in the first ( ).
![x^2(x^2)=x^4](https://tex.z-dn.net/?f=x%5E2%28x%5E2%29%3Dx%5E4)
![x^2(-6x)=-6x^3](https://tex.z-dn.net/?f=x%5E2%28-6x%29%3D-6x%5E3)
![x^2(13)=13x^2](https://tex.z-dn.net/?f=x%5E2%2813%29%3D13x%5E2)
![-6x(x^2)=-6x^3](https://tex.z-dn.net/?f=-6x%28x%5E2%29%3D-6x%5E3)
![-6x(-6x)=36x^2](https://tex.z-dn.net/?f=-6x%28-6x%29%3D36x%5E2)
![-6x(13)=-78x](https://tex.z-dn.net/?f=-6x%2813%29%3D-78x)
![10(x^2)=10x^2](https://tex.z-dn.net/?f=10%28x%5E2%29%3D10x%5E2)
![10(-6x)=-60x](https://tex.z-dn.net/?f=10%28-6x%29%3D-60x)
![10(13)=130](https://tex.z-dn.net/?f=10%2813%29%3D130)
--------------------------------Combine like terms:
![x^4-12x^3+59x^2-138x+130](https://tex.z-dn.net/?f=x%5E4-12x%5E3%2B59x%5E2-138x%2B130)
-----------------------------------------------------------------
So now we have
![(x-2)(x^4-12x^3+59x^2-138x+130](https://tex.z-dn.net/?f=%28x-2%29%28x%5E4-12x%5E3%2B59x%5E2-138x%2B130)
We are almost done.
We are going to multiply the first term in the first ( ) to every term in the second ( ).
We are going to multiply the second term in the first ( ) to every term in the second ( ).
![x(x^4)=x^5](https://tex.z-dn.net/?f=x%28x%5E4%29%3Dx%5E5)
![x(-12x^3)=-12x^4](https://tex.z-dn.net/?f=x%28-12x%5E3%29%3D-12x%5E4)
![x(59x^2)=59x^3](https://tex.z-dn.net/?f=x%2859x%5E2%29%3D59x%5E3)
![x(-138x)=-138x^2](https://tex.z-dn.net/?f=x%28-138x%29%3D-138x%5E2)
![x(130)=130x](https://tex.z-dn.net/?f=x%28130%29%3D130x)
![-2(x^4)=-2x^4](https://tex.z-dn.net/?f=-2%28x%5E4%29%3D-2x%5E4)
![-2(-12x^3)=24x^3](https://tex.z-dn.net/?f=-2%28-12x%5E3%29%3D24x%5E3)
![-2(59x^2)=-118x^2](https://tex.z-dn.net/?f=-2%2859x%5E2%29%3D-118x%5E2)
![-2(-138x)=276x](https://tex.z-dn.net/?f=-2%28-138x%29%3D276x)
![-2(130)=-260](https://tex.z-dn.net/?f=-2%28130%29%3D-260)
----------------------------------------Combine like terms:
![x^5-14x^4+83x^3-256x^2+406x-260](https://tex.z-dn.net/?f=x%5E5-14x%5E4%2B83x%5E3-256x%5E2%2B406x-260)