Problem 67 - Algebra and Trigonometry

Question

Explaining the Concepts. Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Step-by-Step Solution

Answer

Descartes's Rule of Signs is applied by arranging a polynomial in decreasing power order, then counting the number of sign changes among the coefficients. The possible number of positive real roots is the same as the number of sign changes, or less than that by an even number.

Step 1: Principle Understanding

Descartes's Rule of Signs states that the number of positive real zeros in a polynomial function corresponds to the number of changes in the sign of the coefficients, or is less than that by an even number. Zero is neither positive nor negative, so we exclude zeros for this rule.

Step 2: Apply the Rule

To implement Descartes's Rule, follow these steps:\n1- Write down the polynomial in decreasing exponent order.\n2- Count the number of sign changes. Every transition from a positive term to a negative one, or vice versa, counts as a sign change.\n3- The number of positive real zeroes is equal to the number of sign changes, or must be less than that by an even number. For example, if you have 5 sign changes, the number of positive real zeroes is either 5, 3, or 1.

Step 3: Key Note

It's critical to remember that Descartes' Rule of Signs provides a possible number of positive real zeros. But the rule doesn't give the actual number of positive zeros; it only provides possibilities. Its main utility is in limiting potential roots which leads to lesser calculations if you're solving the functions by hand.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression that consists of variables raised to various power degrees, combined using addition, subtraction, and multiplication. Polynomials are fundamental in algebra and can be seen in many mathematical equations.

In a polynomial function, terms are arranged in order of descending exponent value. For example, the polynomial \( 3x^4 - 2x^3 + 7x - 5 \) is arranged from the highest degree (4) to the constant term. Each term consists of a coefficient and a variable raised to a power.

• Degree: This is the highest exponent found in the polynomial's terms. In the example above, the degree is 4.
• Coefficients: The numbers in front of the variable, like 3, -2, 7, and -5, are the coefficients.
• Leading Term/Leading Coefficient: The term with the highest power, 3x^4, is the leading term, and 3 is the leading coefficient.

Understanding how to interpret polynomial functions and rearrange them is essential when applying Descartes's Rule of Signs.

Positive Real Zeros

Positive real zeros of a polynomial function are the values of \( x \) that satisfy the equation \( f(x) = 0 \) and are greater than zero.

In simple terms, these are the real solutions that are positive numbers when we plot the polynomial equation on a graph.

• Roots: These are the solutions or zeros of the function where the polynomial equals zero.
• Positive Zeros: Roots that are greater than zero.

Identifying these zeros helps in understanding the behavior of a polynomial function as they indicate where the graph crosses or touches the x-axis at a point greater than zero.
Using Descartes’s Rule of Signs allows us to predict how many of these positive real zeros we might encounter, even before calculating them accurately.

Sign Changes

Sign changes in a polynomial function are crucial for determining the number of positive real zeros using Descartes's Rule of Signs. A sign change occurs between two successive terms if one of them is positive and the next one is negative, or vice versa.

• For example, in the polynomial \( -3x^4 + 2x^3 - 7x + 5 \), the sign changes are observed between:
  1. \(-3x^4\) (negative) to \(+2x^3\) (positive)
  2. \(+2x^3\) (positive) to \(-7x\) (negative)
  3. \(-7x\) (negative) to \(+5\) (positive)

This counts as three sign changes. According to Descartes's Rule of Signs, the number of positive real zeros could be 3, 1, or possibly none.
This rule provides a systematic approach to predict the possible number of positive roots by simply counting the pattern of changes in the signs of coefficients from one term to the next.

Count Coefficients

Counting coefficients in a polynomial function is an essential step in determining the sign changes for Descartes’s Rule of Signs. Each term in a polynomial consists of a coefficient, which is the numerical part of the term.

To use Descartes's Rule of Signs effectively, one must list the polynomial in descending order and carefully analyze the sequence of coefficients:

• Highlight Sign Transition: Pay attention to where the sign changes from positive to negative or vice versa.
• Maintaining Order: Ensure that all coefficients correspond to the correct power of the variable.
• Exclusion of Zero Coefficients: If a term is missing (or its coefficient is zero), it does not count in sign change calculations.

This count is crucial to applying Descartes's Rule effectively. By ensuring that no sign change is overlooked, one can accurately estimate the possible number of positive real zeros, aiding immensely in solving polynomial equations.