Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.3

Question 2

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Factorize into linear factors.

(v) z416z^4 - 16

Solution

Recognize difference of squares:z416=(z24)(z2+4)Factor each:z24=(z2)(z+2)z2+4=(z2i)(z+2i)Therefore:z416=(z2)(z+2)(z2i)(z+2i)\begin{aligned} & \boxed{\text{Recognize difference of squares:}} \\ \\ & z^4 - 16 = (z^2 - 4)(z^2 + 4) \\ \\ & \boxed{\text{Factor each:}} \\ & z^2 - 4 = (z - 2)(z + 2) \\ & z^2 + 4 = (z - 2i)(z + 2i) \\ \\ & \boxed{\text{Therefore:}} \\ & z^4 - 16 = (z - 2)(z + 2)(z - 2i)(z + 2i) \end{aligned}