List the number of complex zeros, possible number of real zeros, and possible rational roots for the following polynomial function.

Question

y = 3x³ − 7x² + 4

Answer 1

By rational zeros theorem list of possible rational zeros: ±1, ±2, ±4, ±1/3, ±2/3, ±4/3

By Descard's rule of sighns: 2 or 0 postive zeros (2 changes of sign in y(x))

1 or 1 negative zeros ( 1 change of sign in y(-x))

0 or 2 complex congugate zeros

3x³ - 7x² + 4 = 0; 3x³ - 3x²- 4x² + 4 = 0; 3x²(x - 1) - 4(x - 1)(x + 1) = 0; (x - 1)(3x² - 4x - 4) = 0, x = 1,

3x² - 4x - 4 = 0; x = (2 ±√4 + 12)/3 = (2 ± 4)/3; x = 2 or x = -2/3

Answer: solutions set {-2/3, 1, 2]