Respuesta :
Answer:
Standard form: [tex]x^5-14x^4+83x^3-256x^2+406x-260[/tex]
Factored form with complex numbers: [tex](x-2)(x-(3+i))(x-(3-i))(x-(3-2i))(x-(3+2i))[/tex]
Factored form without complex numbers: [tex](x-2)(x^2-6x+10)(x^2-6x+13)[/tex]
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:
[tex](x-2)(x-(3+i))(x-(3-i))(x-(3-2i))(x-(3+2i))[/tex]
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.
[tex](x-(m+ni))(x-(m-ni))[/tex]
I'm going to use foil.
First: [tex]x(x)=x^2[/tex]
Outer: [tex]x(-(m-ni))=-(m-ni)x[/tex]
Inner: [tex]-(m+ni)(x)=-(m+ni)x[/tex]
Last: [tex]-(m+ni)(-(m-ni))=(m+ni)(m-ni)=m^2-n^2i^2=m^2-n^2(-1)=m^2+n^2[/tex]
(Note: [tex](a-b)(a+b)=a^2-b^2[/tex]; When multiplying conjugates, you just have to multiply the first terms of each and the last terms of each.)
------------------------------------Let's combine these terms:
[tex]x^2-(m-ni)x-(m+ni)x+m^2+n^2[/tex]
Distribute:
[tex]x^2-mx+nix-mx-nix+m^2+n^2[/tex]
Combine like terms:
[tex]x^2-2mx+m^2+n^2[/tex]
So the formula we will be using on the factors that contain conjugate pair zeros is:
[tex](x-(m+ni))(x-(m-ni))=x^2-2mx+m^2+n^2[/tex].
[tex](x-(3+i))(x-(3-i))=x^2-2(3)x+3^2+1^2[/tex]
[tex](x-(3+i))(x-(3-i))=x^2-6x+9+1[/tex]
[tex](x-(3+i))(x-(3-i))=x^2-6x+10[/tex]
[tex](x-(3+2i))(x-(3-2i))=x^2-2(3)x+3^2+2^2[/tex]
[tex](x-(3+2i))(x-(3-2i))=x^2-6x+9+4[/tex]
[tex](x-(3+2i))(x-(3-2i))=x^2-6x+13[/tex]
------------------------------------------------
So this is what we have now:
[tex](x-2)(x^2-6x+10)(x^2-6x+13)[/tex]
I'm going to multiply the last two factors:
[tex](x^2-6x+10)(x^2-6x+13)[/tex]
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 ( ).
[tex]x^2(x^2)=x^4[/tex]
[tex]x^2(-6x)=-6x^3[/tex]
[tex]x^2(13)=13x^2[/tex]
[tex]-6x(x^2)=-6x^3[/tex]
[tex]-6x(-6x)=36x^2[/tex]
[tex]-6x(13)=-78x[/tex]
[tex]10(x^2)=10x^2[/tex]
[tex]10(-6x)=-60x[/tex]
[tex]10(13)=130[/tex]
--------------------------------Combine like terms:
[tex]x^4-12x^3+59x^2-138x+130[/tex]
-----------------------------------------------------------------
So now we have
[tex](x-2)(x^4-12x^3+59x^2-138x+130[/tex]
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 ( ).
[tex]x(x^4)=x^5[/tex]
[tex]x(-12x^3)=-12x^4[/tex]
[tex]x(59x^2)=59x^3[/tex]
[tex]x(-138x)=-138x^2[/tex]
[tex]x(130)=130x[/tex]
[tex]-2(x^4)=-2x^4[/tex]
[tex]-2(-12x^3)=24x^3[/tex]
[tex]-2(59x^2)=-118x^2[/tex]
[tex]-2(-138x)=276x[/tex]
[tex]-2(130)=-260[/tex]
----------------------------------------Combine like terms:
[tex]x^5-14x^4+83x^3-256x^2+406x-260[/tex]
