Problem 35 - Algebra and Trigonometry

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. $$ f(x)=5 x^{3}-3 x^{2}+3 x-1 $$

Step-by-Step Solution

Answer

The given function \(f(x)=5 x^{3}-3 x^{2}+3 x-1\) can have 1 or no positive real zeros and 2 or no negative real zeros.

Step 1: Determine the possible number of positive real zeros

First, express the function in standard form \(f(x)=5 x^{3}-3 x^{2}+3 x-1\). Now, count the number of sign changes. A sign change occurs every time consecutive coefficients of the terms are of different signs. In this case, there is one sign change from \(5x^3\) to \(-3x^2\). So, there could be 1 or (1-2)=0 positive real root.

Step 2: Determine the possible number of negative real zeros

Now, consider \(f(-x)=5(-x)^{3}-3(-x)^{2}+3(-x)-1 = -5x^{3}-3x^{2}-3x-1\). Again, find the number of sign changes. There is one sign change from \(-5x^3\) to \(-3x^2\) and another from \(-3x^2\) to \(-3x\). So, there could be 2 or (2-2)=0 negative real roots.

Step 3: Summarize the results

Based on Descartes’s Rule of Signs, this function can have 1 or 0 positive real zeros and also can have 2 or 0 negative real zeros.

Key Concepts

Real Zeros of a Polynomial

Understanding the real zeros of a polynomial is pivotal for graphing functions and solving polynomial equations. Often referred to as roots or x-intercepts, the real zeros are the x-values where the polynomial equals zero. In other words, if you substitute a zero back into the polynomial, the output should be zero. This property is used to determine various characteristics of the function, including its behavior and graph.

For example, consider the polynomial function given by the equation \( f(x) = 5x^{3} - 3x^{2} + 3x - 1 \). The real zeros are the solutions to the equation \( 5x^{3} - 3x^{2} + 3x - 1 = 0 \). These zeros are critical as they are the points where the graph of the polynomial will cross or touch the x-axis. Calculating or estimating these values involves factoring the polynomial (if possible), using numerical methods, or applying theorems like Descartes's Rule of Signs.

Polynomial Sign Changes

The concept of polynomial sign changes is directly tied to Descartes's Rule of Signs, which is a handy tool for predicting the number of positive and negative real roots of a polynomial without actually solving the equation. A sign change in a polynomial occurs whenever consecutive non-zero coefficients differ in sign.

Take, for example, the polynomial \( f(x) = 5x^{3} - 3x^{2} + 3x - 1 \). The sign of the coefficients changes from positive to negative, and then from negative to positive. Counting these changes can tell us possible numbers of positive real zeros. In this case, there is one sign change, which suggests we can expect exactly one or three fewer than this – hence, zero positive real roots. This counting process is the first step in applying Descartes's Rule of Signs, and though it does not give us the exact number of roots, it narrows down the possibilities significantly.

It's important to remember that this rule considers only the possibility of real zeros and doesn't account for complex roots which may also exist in the polynomial.

Positive and Negative Real Roots

The positive and negative real roots of a polynomial can be inferred using Descartes's Rule of Signs, which has two parts. To find the number of positive real roots, you apply the rule to the polynomial as is. To determine the possible negative real roots, you modify the polynomial by replacing every occurrence of 'x' with '-x' and then apply the rule.

As per the rule, the potential number of positive real roots is equal to the number of sign changes in the polynomial's coefficients or less than that by an even number. For negative real roots, we look for sign changes in the modified polynomial. The exercise provided exhibits this beautifully: \(f(-x) = -5x^{3} - 3x^{2} - 3x - 1\), and upon examination, it's clear that it has two sign changes indicating that there could be either two or zero negative real roots.

Remember, this rule gives the maximum number of positive or negative real roots, not the exact amount. This means that a polynomial with two sign changes can have two, zero, but never three or one negative real roots. There can be fewer real roots than the maximum suggested by Descartes due to the presence of complex roots, which do not change the sign of the function when graphed over the real numbers.