Question

For each of the following functions, (a) Use Descartes' Rule of Sign to determine all the possible positive-negative-complex configurations of the zeros. (b) Assuming all the zeros are real, determine (if possible) an interval containing them. (c) Determine all 5 zeros. 1. f(x)=xFor each of the following functions, (a) Use Descartes' Rule of Sign to determine all the possible positive-negative-complex configurations of the zeros. (b) Assuming all the zeros are real, determine (if possible) an interval containing them. (c) Determine all 5 zeros.

1. f(x)=x⁵−3x⁴−22x³ +74x² −75x+25
2. f(x)=x⁵ −4x⁴+x³ +10x² −4x−8
3. f(x)=x⁵⁵−6x⁴ +58x³
4. f(x)=x⁵+x³3−2x−12x²−8

Answer 1

Descartes' Rule of Signs can predict the number of positive and negative real zeros for the given polynomials; the first function might have 3 or 1 positive real zeros and 0 negative real zeros. Determining the exact zeros would require additional methods and cannot be confirmed without deeper analysis or the use of specific tools like synthetic division or numerical solvers.

Step-by-step explanation:

Regarding the functions presented, we can utilize Descartes' Rule of Signs to predict the number of positive and negative real zeros. The Rule of Signs states that the number of positive real zeros of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients, or is less than this by an even number. The number of negative real zeros can be determined in a similar way by evaluating the polynomial with x replaced by -x.

For the first function f(x) = x⁵ - 3x⁴ - 22x³ + 74x² - 75x + 25, there are 3 sign changes, indicating there could be 3 or 1 positive real zeros. For negative real zeros, we replace x with -x and notice there are no sign changes, suggesting 0 negative real zeros. All remaining zeros would be complex, with real and imaginary parts.

Assuming all zeros are real for a polynomial of degree 5 is a complex task without specific numerical methods or graphical tools. For example, an interval for the first polynomial's zeros cannot be easily determined without further analysis or graphing.

To determine the zeros, we would typically use methods such as synthetic division, factoring, the Rational Root Theorem, or numerical solvers to find the exact zeros.