Question

State the number of complex roots, the number of positive real roots and the number of negative real roots of

x^4-x^3-6x+ 3 =0

Answer 1

9514 1404 393

Answer:

• 2 complex roots
• 2 positive real roots
• 0 negative real roots

Explanation:

The signs of the terms are + - - +. There are two sign changes, so 0 or 2 positive real roots.

Negating the signs of the odd-degree terms, the signs are + + + +. There are no sign changes, so 0 negative real roots.

For x=0, the value of the quartic is +3. For x=1, the value is -3, so we know there are 2 positive real roots, one of which lies in the interval (0, 1).

The 4th-degree polynomial equation must have 4 roots, so the other two must be complex.

• 2 complex roots
• 2 positive real roots
• 0 negative real roots

_____

The roots are approximately 0.489999841592, 2.06573034434, −0.777865092969 ± 1.53582061225i