Problem 25 - Precalculus

Question

Use Descartes's rule of signs to obtain information regarding the roots of the equations. $$x^{3}+5=0$$

Step-by-Step Solution

Answer

No positive real roots; exactly one negative real root.

Step 1: Convert to Standard Form

The given equation is already in a polynomial form: \(x^3 + 5 = 0\). To use Descartes’s Rule of Signs, express it as \(x^3 + 0x^2 + 0x + 5 = 0\). This way, the polynomial function is \(f(x) = x^3 + 0x^2 + 0x + 5\).

Step 2: Identify Sign Changes in the Polynomial

Observe the coefficients of the polynomial \(f(x) = x^3 + 0x^2 + 0x + 5\). The coefficients are \(1, 0, 0, 5\). The sign changes from positive to positive (as 0 does not affect changes), so there are no sign changes in \(f(x)\). According to Descartes's rule, the number of positive real roots is zero.

Step 3: Analyze Roots for Negative Values

To analyze negative roots, consider \(f(-x)\):\(-x^3 + 0x^2 + 0x + 5\). The coefficients are \(-1, 0, 0, 5\), so the signs change from negative to positive once at the transition from \(-1\) to \(5\). So, according to Descartes's Rule, there is one negative real root.

Key Concepts

polynomial equations

Polynomial equations are mathematical expressions that consist of variables raised to whole number exponents and have one or more terms. The form of these equations can range from the simplest to highly complex structures.
In our example, the given equation is a polynomial equation of degree 3, expressed as:

• \(x^3 + 5 = 0\)

To use various mathematical rules, such as Descartes's Rule of Signs, we often express the polynomial in standard form, ensuring all terms are properly accounted. The standardized form of this particular polynomial would look like:

• \(f(x) = x^3 + 0x^2 + 0x + 5\)

Each coefficient in a polynomial equation can affect the way we analyze its roots, as different tools and rules are based on reading these coefficients and their signs. Understanding the structure of a polynomial helps in determining the behavior and roots of the equation.

positive and negative roots

The roots of a polynomial equation are the solutions that make the equation equal to zero. Descartes's Rule of Signs can predict the number of possible positive and negative roots based on the sign changes in the polynomial.When assessing for positive roots, we look directly at the polynomial as is. In the case of:

• \(f(x) = x^3 + 0x^2 + 0x + 5\)

There are no sign changes in its coefficients (all being non-negative), suggesting zero positive roots.To explore negative roots, we substitute \(-x\) for \(x\), examining the new polynomial:

• \(f(-x) = -x^3 + 0x^2 + 0x + 5\)

This reveals one sign change, indicating one negative root. These insights into positive and negative roots allow us to predict how many solutions the equation could have.

sign changes in polynomials

Sign changes in polynomials are examined to determine the potential number of real roots in different sign categories (positive or negative). This approach stems from Descartes's Rule of Signs.For our polynomial \(f(x) = x^3 + 0x^2 + 0x + 5\), when you observe the non-zero coefficients \([1, 0, 0, 5]\) for \(x\) values and \([-1, 0, 0, 5]\) for \(-x\), your focus is on how often the signs change between these coefficients.
- With \(f(x)\), there is:

• No change from positive to negative or vice versa, so no positive roots.

- By calculating \(f(-x)\), you spot:

• One change from negative to positive.

Sign changes are trick indicators that help predict if and how the roots manifest, crucial to solving polynomials and finding possible real solutions.