Question 178605
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*[tex \Large x^4 - 4096 = 0]


First thing to notice is that you have the difference of two squares:



*[tex \Large (x^2)^2 - 64^2 = 0], so you can factor:


*[tex \Large (x^2 + 64)(x^2 - 64) = 0]


Using the Zero Product Rule:


*[tex \Large x^2 - 64 = 0], or


*[tex \Large x^2 + 64 = 0]


If *[tex \Large x^2 - 64 = 0], then we can again factor the difference of two squares:


*[tex \Large (x + 8)(x - 8) = 0] and we have our two real roots, namely 8 and -8.


On the other hand, if *[tex \Large x^2 + 64 = 0], then *[tex \Large x^2 = -64] so *[tex \Large x = \pm i sqrt{64}] which is to say *[tex \Large x = \pm 8i] giving us our conjugate pair of complex roots.


In order, your answers should be:


*[tex \Large \text {          } x_{rl} = 8]


*[tex \Large \text {          } x_{rs}= -8]


*[tex \Large \text {          } x_{il} = 8i]


*[tex \Large \text {          } x_{is}= -8i]





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