Question

Match each expression to its equivalent standard form.

Answer 1

(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

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 2

1. The first expression is equivalent to
`x^2+2x+2`.

2. The second expression is equivalent to
`x^2+4`.

3. The third expression is equivalent to
`x^2-4x+8`.

Explanation:

(1).

The given expression is

`(x+1+i)(x+1-i)`

`[(x+1)+i][(x+1)-i]`

Using the algebraic properties, we get

`(x+1)^2-(i)^2`
`[\because a^2-b^2=(a-b)(a+b)]`

`x^2+2x+1-(i)^2`
`[\because (a+b)^2=a^2+2ab+b^2]`

`x^2+2x+1-(-1)`
`[\because i^2=-1]`

`x^2+2x+2`

Therefore the first expression is equivalent to
`x^2+2x+2`.

(2).

The given expression is

`(x+2i)(x-2i)`

Using the algebraic properties, we get

`(x)^2-(2i)^2`
`[\because a^2-b^2=(a-b)(a+b)]`

`x^2-4i^2`

`x^2-4(-1)`
`[\because i^2=-1][/tex</p><p>[tex]x^2+4`

Therefore the second expression is equivalent to

`x^2+4`.

(3)

The given expression is

`(x-2+2i)(x-2-2i)`

`[(x-2)+2i][(x-2)-2i]`

Using the algebraic properties, we get

`(x-2)^2-(2i)^2`
`[\because a^2-b^2=(a-b)(a+b)]`

`x^2-4x+4-4i^2`
`[\because (a-b)^2=a^2-2ab+b^2]`

`x^2-4x+4-4(-1)`
`[\because i^2=-1]`

`x^2-4x+4+4`

`x^2-4x+8`

Therefore the third expression is equivalent to
`x^2-4x+8`.