Question

Please help Make a table showing the possible numbers of positive real zeros, negative

real zeros, and imaginary zeros for the function

Answer 1

f(x) can have 0-5 real zeros and 0-5 imaginary zeros, adding up to 5 total zeros (max).

Sure, I can help you with that. Here is a table showing the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function
`f(x) = 2x^5 - 5x^3 + 2x - 6:`

Number of Positive Real Zeros Number of Negative Real Zeros Number of Imaginary Zeros Total Number of Zeros

0, 1, 2, 3, 4, or 5 0, 1, 2, 3, 4, or 5 0, 1, 2, 3, 4, or 5 5

The number of possible positive real zeros is determined by Descartes' rule of signs, which states that the number of positive real zeros of a polynomial is equal to the number of changes in sign between the coefficients of the polynomial, or is less than that number by an even number.

The number of possible negative real zeros is determined by the same rule, but applied to the polynomial -f(-x).

The number of possible imaginary zeros is determined by the complex conjugate root theorem, which states that if a polynomial with real coefficients has a nonreal complex root, then its complex conjugate is also a root.

For example, if
`f(x) = 2x^2 - 5x + 2` has a root of 1 + 2i, then it must also have a root of 1 - 2i.

The total number of zeros of a polynomial is equal to its degree.

In the case of
`f(x) = 2x^5 - 5x^3 + 2x - 6`, the degree is 5, so there can be a maximum of 5 zeros.