Question

Factor over complex numbers 2x^4+36x^2+162

Answer 1

Factor over complex numbers of
`2 x^(4)+36 x^(2)+162` is
`2(x+3 i)^(2)(x-3 i)^(2)`

Solution:

From question given that

`2 x^(4)+36 x^(2)+162` → (1)

On substituting
`x^(2)=y` in equation (1),

`=2 y^(2)+36 y+162`

Taking 2 as a common in above expression,

`=2\left(y^(2)+18 y+81\right)`

Rewrite the above expression,

`=2\left[y^(2)+2(y)(9)+9^(2)\right]`

`=2(y+9)^(2)`
`\left[\text { Using }(a+b)^(2)=a^(2)+2 a b+b^(2)\right]`

`=2\left(x^(2)+9\right)^(2)`
`\left[\text { since } y=x^(2)\right]`

`\left[\text { Using } a^(2)+b^(2)=(a+i b)(a-i b), \text { where } i=√(-1)\right]`

`=2[(x+3 i)(x-3 i)]^(2)`

`=2(x+3 i)^(2)(x-3 i)^(2)`

Hence Factor over complex numbers of
`2 x^(4)+36 x^(2)+162` is
`2(x+3 i)^(2)(x-3 i)^(2)`