Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $f\left(x\right)=3x^4-2x^3-8x+5$

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1. Write down the polynomial function: \(f(x) = 3x^4 - 2x^3 - 8x + 5\).
2. To find the possible number of positive real zeros, count the number of sign changes in the coefficients of \(f(x)\). The coefficients are \$3\(, \)-2\(, \)0\( (for \)x^2\( term), \)-8\(, and \)5$. Note that the zero coefficient does not affect sign changes.
3. Identify the sign changes in \(f(x)\): from \$3\( to \)-2\( (positive to negative), from \)-2\( to \)0\( (no sign change since zero is neutral), from \)0\( to \)-8\( (no sign change), and from \)-8\( to \)5$ (negative to positive). Count these sign changes to determine the possible number of positive real zeros.
4. To find the possible number of negative real zeros, evaluate \(f(-x)\) by substituting \(-x\) into the function: \(f(-x) = 3(-x)^4 - 2(-x)^3 - 8(-x) + 5\). Simplify this expression to get a new polynomial.
5. Count the number of sign changes in the coefficients of \(f(-x)\) to determine the possible number of negative real zeros. According to Descartes's Rule of Signs, the number of positive or negative real zeros is either equal to the number of sign changes or less than that by an even number.

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a method used to determine the possible number of positive and negative real zeros of a polynomial function by counting the sign changes in the coefficients. The number of positive real zeros is equal to the number of sign changes in f(x) or less than that by an even number. For negative zeros, the rule is applied to f(-x).

Polynomial Functions and Their Zeros

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. The zeros of a polynomial are the values of x that make the function equal to zero. Understanding the degree and terms of the polynomial helps in analyzing the possible number and nature of its zeros.

Evaluating f(-x) for Negative Zeros

To find the possible number of negative real zeros using Descartes's Rule of Signs, substitute -x into the polynomial to get f(-x). This changes the signs of terms with odd powers of x. Counting the sign changes in f(-x) then gives the possible number of negative real zeros, similar to the process for positive zeros.