Solve each problem. Use Descartes' rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of ƒ(x)=x^3+3x^2-4x-2.

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1. Identify the polynomial function: \( f(x) = x^3 + 3x^2 - 4x - 2 \).
2. To find the number of positive real zeros, count the sign changes in \( f(x) \). The sequence of coefficients is \( 1, 3, -4, -2 \), which changes signs twice (from 3 to -4 and from -4 to -2).
3. According to Descartes' Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even number. Therefore, there can be 2 or 0 positive real zeros.
4. To find the number of negative real zeros, consider \( f(-x) \). Substitute \(-x\) into the polynomial: \( f(-x) = (-x)^3 + 3(-x)^2 - 4(-x) - 2 = -x^3 + 3x^2 + 4x - 2 \).
5. Count the sign changes in \( f(-x) \). The sequence of coefficients is \(-1, 3, 4, -2 \), which changes signs twice (from -1 to 3 and from 4 to -2). Therefore, there can be 2 or 0 negative real zeros.

Key Concepts

Descartes' Rule of Signs

Descartes' 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 based on the number of sign changes in its coefficients. For positive roots, you count the sign changes in the polynomial as it is, while for negative roots, you evaluate the polynomial at -x and count the sign changes in that expression. This rule helps narrow down the possible number of real roots before further analysis.

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 is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the degree of the polynomial and its leading coefficient is crucial for analyzing its behavior and the number of roots it can have.

Complex Zeros

Complex zeros of a polynomial are solutions to the equation f(x) = 0 that are not real numbers. They occur in conjugate pairs due to the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots in the complex number system, counting multiplicities. Recognizing the presence of complex zeros is essential for fully understanding the behavior of polynomial functions, especially when real roots are limited.