Respuesta :
Answer:
(x+1+i)(x+1-i) goes with x^2+2x+2
(x+2i)(x-2i) goes with x^2+4
(x-2+2i)(x-2-2i) goes with x^2-4x+8
Step-by-step explanation:
(x+1+i)(x+1-i)
(x+[1+i])(x+[1-i])
Use foil.
First: x(x)=x^2
Outer: x(1+i)=x+ix
Inner: x(1-i)=x-ix
Last: (1+i)(1-i)=1-i^2 since 1+i and 1-i are conjugates
__Add together to get: x^2+2x+1-i^2
We can actually simplify this because i^2=-1
So x^2+2x+1-i^2=x^2+2x+1-(-1)=x^2+2x+2
(x+2i)(x-2i)
These are conjugates so just do first and last of foil.
First: x(x)=x^2
Last: 2i(-2i)=-4i^2=-4(-1)=4
==Adding together gives x^2+4
(x-2+2i)(x-2-2i)
(x+[-2+2i])(x+[-2-2i])
This is similar to first.
Foil time!
First: x(x)=x^2
Outer: x(-2-2i)=-2x-2ix
Inner: x(-2+2i)=-2x+2ix
Last: (-2-2i)(-2+2i)=4-4i^2 (multiplying conjugates again)
==Add together giving us x^2-4x+4-4i^2
This can be simplified since i^2=-1.
So applying this gives us x^2-4x+4-4(-1)
=x^2-4x+4+4
=x^2-4x+8
Answer:
1. The first expression is equivalent to [tex]x^2+2x+2[/tex].
2. The second expression is equivalent to [tex]x^2+4[/tex].
3. The third expression is equivalent to [tex]x^2-4x+8[/tex].
Step-by-step explanation:
(1).
The given expression is
[tex](x+1+i)(x+1-i)[/tex]
[tex][(x+1)+i][(x+1)-i][/tex]
Using the algebraic properties, we get
[tex](x+1)^2-(i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2+2x+1-(i)^2[/tex] [tex][\because (a+b)^2=a^2+2ab+b^2][/tex]
[tex]x^2+2x+1-(-1)[/tex] [tex][\because i^2=-1][/tex]
[tex]x^2+2x+2[/tex]
Therefore the first expression is equivalent to [tex]x^2+2x+2[/tex].
(2).
The given expression is
[tex](x+2i)(x-2i)[/tex]
Using the algebraic properties, we get
[tex](x)^2-(2i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2-4i^2[/tex]
[tex]x^2-4(-1)[/tex] [tex][\because i^2=-1][/tex
[tex]x^2+4[/tex]
Therefore the second expression is equivalent to
[tex]x^2+4[/tex].
(3)
The given expression is
[tex](x-2+2i)(x-2-2i)[/tex]
[tex][(x-2)+2i][(x-2)-2i][/tex]
Using the algebraic properties, we get
[tex](x-2)^2-(2i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2-4x+4-4i^2[/tex] [tex][\because (a-b)^2=a^2-2ab+b^2][/tex]
[tex]x^2-4x+4-4(-1)[/tex] [tex][\because i^2=-1][/tex]
[tex]x^2-4x+4+4[/tex]
[tex]x^2-4x+8[/tex]
Therefore the third expression is equivalent to [tex]x^2-4x+8[/tex].
