In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x^4−5x^3−x^2−6x+4

Verified step by step guidance

1. Identify the polynomial function: $f(x)=2x^4-5x^3-x^2-6x+4$.
2. To find the possible number of positive real zeros, examine the sign changes in $f(x)$.
3. List the coefficients of $f(x)$: $2,-5,-1,-6,4$.
4. Count the sign changes between consecutive coefficients: $2\to -5$ (change), $-5\to -1$ (no change), $-1\to -6$ (no change), $-6\to 4$ (change).
5. There are 2 sign changes, so there can be 2 or 0 positive real zeros.

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a mathematical theorem that provides a way to determine the number of positive and negative real roots of a polynomial function. It 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 less than that by an even integer. Similarly, for negative real zeros, one evaluates the polynomial at -x and counts the sign changes in that expression.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable x is given by f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n is not zero. Understanding the degree and leading coefficient of the polynomial is essential for applying Descartes's Rule of Signs effectively.

Sign Changes

Sign changes refer to the transitions between positive and negative values in a sequence of numbers. In the context of Descartes's Rule of Signs, identifying sign changes in the coefficients of a polynomial helps determine the potential number of real roots. For example, if a polynomial has coefficients of alternating signs, it indicates a higher likelihood of real zeros, which is crucial for analyzing the function's behavior.