Question

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

Answer 1

Answer:

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,

$=2\left[y^{2}+2(y)(9)+9^{2}\right]$

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

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

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

$=2[(x+3 i)(x-3 i)]^{2}$

$=2(x+3 i)^{2}(x-3 i)^{2}$

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