Factored form of the given polynomial function over the complex numbers will be f(x) = (x - 1)(x + 1)(x - 2i)(x + 2i)
Given polynomial function is,
f(x) = x⁴ + 3x² - 4
Factor this polynomial.
f(x) = x⁴ + 4x² - x² - 4
= x²(x² + 4) - 1(x² + 4)
= (x² - 1)(x² + 4)
= (x - 1)(x + 1)(x² + 4) [Since, (x² - 1) = (x - 1)(x + 1)]
= (x - 1)(x + 1)(x - 2i)(x + 2i) [Since, (x² + 4) = (x - 2i)(x + 2i)]
Therefore, factored form of the given polynomial function 'f' will be
f(x) = (x - 1)(x + 1)(x - 2i)(x + 2i)
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