Question

What are the possible numbers of positive, negative, and complex zeros of f(x) = -x^6 + x^5- x^4 + 4x^3 - 12x^2 + 12 ?


Positive: 4, 2, or 0; negative: 2 or 0; complex: 6, 4, 2, or 0
Positive: 3 or 1; negative: 3 or 1; complex: 4, 2, or 0
Positive: 2 or 0; negative: 2 or 0; complex: 6, 4, or 2
Positive: 5, 3, or 1; negative: 1; complex: 4, 2, or 0

Answer 1

Have you studied Descartes' Rule of Signs yet? @Everly

Answer 2

Yes but not too sure if I grasp it 100%

Answer 3

Okay. I'll attach the statement of the rule and also a sample worked problem.

Finding the roots of polynomials when the polynomials will not factor becomes difficult at times but we can get some idea about the roots from Descartes.

Answer 4

Two attachments.

Answer 5

First up, we need to look for the number of sign changes here:

f(x) = -x^6 + x^5- x^4 + 4x^3 - 12x^2 + 12

That will give some idea about the number of positive real roots that this polynomial MAY have.

Answer 6

@Everly Okay so far?

Answer 7

Yes, everything seems straight-forward for now

Answer 8

5 sign changes, right?

Answer 9

Look for sign changes:

There are 5 (count them).

The number of real positive real roots will be 5 or less than 5 by a multiple of 2.

That means that there may be 5 positive real roots, or 3 positive real roots or 1 positive real root.