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?

Answer 1

The number of zeros are; Positive; 2 or 0; Negative: 4 or 2 or 0; Complex; 0, 2, 4, 6  

How to find the zeros of a Polynomial?

We are given the polynomial function as;

f(x) = x⁶ + x⁵ + x⁴ + 4x³ - 12x² + 12

Use Descartes Rule rule of signs allows us to determine the possible number of positive, negative, and complex roots.

For the function f(x), the number of sign changes that occur from term to term will give us the number of positive possible roots.

For the function f(-x), the number of sign changes that occur from term to term will give us the number of negative possible roots.

The numbers of complex roots is the difference between the degree of function and total number of positive and negative roots.

For f(x), the signs change 2 times.  Thus, there can only be a possibility of 2 positive roots

For f(-x), we have;

x⁶ - x⁵ + x⁴ - 4x³ - 12x² + 12

Thus, the sign changes 4 times and so we have a possibility of 4 negative roots.

Read more about Descartes rule of polynomials at; brainly.com/question/12006853

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