One question about polynomial functions

Question

1. For the quartic functions, ie y= ax^4 + bx^3 + cx^2 + dx + e, state all the possible types and number of roots one could get.

Answer 1

Use Descartes rule.

 

To find the number of positive zeros, we evaluate f(x) and see the number of sign changes.

To find the number of negative zeros, we evaluate f(-x) and see the number of sign changes.

 

f(x) has no sign change

 

f(-x) = ax⁴ - bx³ + cx² - dx + e

 

f(-x) has 4 sign changes

 

 

Here is your possibilities in a chart:

 

# of positive zero |  # of negative zeros  |  # of complex zeros

_____________________________________________________

       0                                4                                 0

       0                                2                                 2