Problem 36 - Algebra for College Students

Question

For Problems \(31-40\), use Descartes' rule of signs (Property 9.6) to help list the possibilities for the nature of the solutions for each equation. \(D o\) not solve the equations. $$ 4 x^{3}+5 x^{2}-6 x-2=0 $$

Step-by-Step Solution

Answer

1 positive root and either 2 or 0 negative roots.

Step 1: Determine the Sign Changes in Polynomial

To use Descartes' rule of signs, first identify the polynomial: \( 4x^{3} + 5x^{2} - 6x - 2 \). Write down the coefficients in order: \(4, 5, -6, -2\). Now, count the number of sign changes: from \(4\) to \(5\) (no change), from \(5\) to \(-6\) (one change), and from \(-6\) to \(-2\) (no change). Therefore, there is one sign change in the polynomial.

Step 2: Determine Possible Positive Real Roots

According to Descartes' rule of signs, the number of positive real roots of a polynomial equation is equal to the number of sign changes, or less than that by an even number. In this case, with one sign change, there can be 1 or \(1 - 2 = -1\), which is invalid. Therefore, there can only be 1 positive real root.

Step 3: Transform and Determine Positive Roots for Negative Roots

To find the possible negative real roots, substitute \(-x\) for \(x\) in the polynomial to get: \(4(-x)^3 + 5(-x)^2 - 6(-x) - 2\). This simplifies to \(-4x^3 + 5x^2 + 6x - 2\). We re-examine the sign changes: from \(-4\) to \(5\) (one change), from \(5\) to \(6\) (zero change), and from \(6\) to \(-2\) (two changes). Thus, there are two sign changes.

Step 4: Determine Possible Negative Real Roots

Using Descartes' rule of signs again, the number of negative real roots is 2 or 0 (as the number of sign changes minus an even number). Thus, there can be 2 or 0 negative real roots.

Key Concepts

Sign Changes in Polynomials

A polynomial is an expression consisting of variables and coefficients, involving terms connected by addition, subtraction, multiplication, and only non-negative integer exponents. Observing sign changes in polynomials is crucial for analyzing their roots using Descartes' rule of signs, a powerful tool that predicts the number of real roots a polynomial may have.
- Start by examining the polynomial and listing its coefficients in order. For the polynomial given, these are: 4, 5, -6, and -2.

• Count each time the sign of the coefficient changes as you move from one term to the next.
• In our example, from 4 to 5 (no change), 5 to -6 (one change), and -6 to -2 (no change).

These sign changes help us determine the number of potential positive and negative real roots, as dictated by Descartes' rule.

Positive Real Roots

In regard to positive real roots, Descartes' rule states that the number of such roots in a polynomial equation is defined by the number of sign changes occurring among its coefficients. This number can be reduced by any even number, including zero.
For instance, our example polynomial shows only one sign change. This indicates that there can be either 1 positive real root or none at all (subtracting 2, an even number, but that would be illogical in this case).
Understanding positive roots helps in determining the behavior and potential solutions of polynomial equations. Comprehension of this concept simplifies assessing whether further analysis or graphing might be desired.

Negative Real Roots

Determining negative real roots involves a similar process to finding positive real roots, but with an initial transformation. Here, substitute (-x) for x in the polynomial and then assess the sign changes in the new polynomial.
For the transformation of the given polynomial: - The substitution results in (-4x^3 + 5x^2 + 6x - 2).

• Inspecting the coefficients: -4 to 5 (one change), 5 to 6 (no change), and 6 to -2 (one change), yielding a total of two sign changes.

The number of negative real roots can then be 2 or decreased by an even number, arriving at 2 or 0 potential roots. This method is vital for a comprehensive grasp of the polynomial's behavior, offering insight beyond simple graphing or estimation methods.