Problem 33 - College Algebra

Question

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. $$ f(x)-x^{3}+2 x^{2}+5 x+4 $$

Step-by-Step Solution

Answer

The function may have one positive real zero and one negative real zero.

Step 1: Identify coefficients

First, the coefficients of the function \(f(x) = -x^{3} + 2x^{2} + 5x + 4\) need to be identified. They are -1, 2, 5, and 4 respectively.

Step 2: Determine sign changes for positive real zeros

To find the number of possible positive real zeros, check the sign changes in the list of coefficients. If a positive coefficient follows a negative one or vice versa, consider it as one sign change. In this case, we go from -1 to 2, resulting in one sign change. Hence, the function could have one positive real zero.

Step 3: Determine sign changes for negative real zeros

To find the number of possible negative real zeros, first substitute \(x\) with \(-x\) in the equation and then do the same as in previous step. The equation becomes \(f(-x) = -x^{3} - 2x^{2} + 5x + 4\). The coefficients are now -1, -2, 5, and 4. This again has one sign change (from -2 to 5), corresponding to possibly one negative real zero.

Key Concepts

Understanding Real Zeros

In mathematics, real zeros of a function are the values of the variable that make the entire function equal to zero. For polynomial functions, these are the points where the graph of the function touches or crosses the x-axis. Identifying the real zeros of a polynomial is a crucial step in solving polynomial equations.

There are two types of zeros to consider:

• **Positive Real Zeros**: These are the x-values greater than zero where the function equals zero.
• **Negative Real Zeros**: These are the x-values less than zero that make the function zero.

Finding real zeros is important because it helps in sketching the graph and understanding the behavior of polynomial functions. Descartes's Rule of Signs can be particularly helpful as it provides a method to predict the possible number of positive and negative real zeros by analyzing the signs of the coefficients.

What Are Polynomial Functions?

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They have the general form:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]where:

• "\(a_n, a_{n-1}, \ldots, a_0\)" are constant coefficients.
• "\(x\)" is the variable.
• "\(n\)" is a non-negative integer, referred to as the degree of the polynomial.

A polynomial function’s behavior and properties, such as the number of zeros, rely significantly on these coefficients and their powers. In the example function \(f(x) = -x^3 + 2x^2 + 5x + 4\), we have:

• Degree: 3 (cubic polynomial)
• Coefficients: -1, 2, 5, and 4

The highest degree of the polynomial dictates its leading term, which usually influences the end behavior of its graph.

Detecting Sign Changes

Sign changes in a sequence of numbers, like a polynomial's coefficients, happen when the sign of the coefficients changes from positive to negative or vice versa. Descartes's Rule of Signs utilizes this concept to estimate the number of possible real zeros.For instance, examine the polynomial \(f(x) = -x^3 + 2x^2 + 5x + 4\):

• The coefficients are \(-1, 2, 5, 4\).
• From \(-1\) to \(2\) is a sign change.

For negative zeros, substitute \(x\) with \(-x\) and apply the same process. For our polynomial,

• Transform it to \(f(-x) = -x^3 - 2x^2 + 5x + 4\).
• The coefficients change to \(-1, -2, 5, 4\).
• There is one sign change, from \(-2\) to \(5\).

Each sign change corresponds to a potential real zero according to Descartes's Rule of Signs. This method efficiently predicts zero possibilities without finding the zeros explicitly.