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german
3 years ago
11

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

Mathematics
1 answer:
Darina [25.2K]3 years ago
7 0

<u>Answer:</u>

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

<u>Solution:  </u>

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}

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