Question: Consider the following polynomial. \[ f(x) = x^3 - 7x^2 + 2x + 40 \] Step 1 of 3: Use…

Consider the following polynomial.

\[ f(x) = x^3 - 7x^2 + 2x + 40 \]

Step 1 of 3: Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros. Be sure to include all possibilities.

Answer

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Separate multiple answers with commas.

Number of Positive Real Zeros: _____

Number of Negative Real Zeros: _____

Solution

Step 1 Consider the polynomial function: \[ f(x) = x^3 - 7x^2 + 2x + 40 \] Step 2 Count the number of sign changes in \( f(x) \): Sign sequence: \( +, -, +, + \) Number of sign changes: 2 Therefore, the possible number of positive real zeros is 2 or 0. Step 3 Evaluate \( f(-x) \): \[ f(-x) = -x^3 - 7x^2 - 2x + 40 \] Count the number of sign changes in \( f(-x) \): Sign sequence: \( -, -, -, + \) Number of sign changes: 1 Thus, the possible number of negative real zeros is 1.