For the function y=x5−5x3−14x, find all real zeros.

Answer 1

Let y = X⁵ – 5X³ – 14X.  Factor out X:

y = X (X4 – 5X² – 14) Let z = X². Then X⁴ – 5X² – 14 = z² – 5z -14 = (z – 7) (z+2)

Therefore y = X (X² – 7) (X² + 2)

Real zeros come X = 0 or X² – 7 = 0; since X² + 2 = 0 has n0 solutions:

X = 0, X = ±  = All real zeros: -, 0,

Answer 2

y = x(x⁴ - 5x² - 14)

y = x(x² - 7)(x² + 2)

x = 0 , ± √7