Problem 48 - College Algebra

Question

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination. \(f(x)=x^{3}-2 x^{2}-5 x+6\)

Step-by-Step Solution

Answer

There is 1 positive root and 1 negative root.

Step 1: Identify the number of sign changes for positive roots

Examine the coefficients of the polynomial \(f(x) = x^3 - 2x^2 - 5x + 6\). The signs of the coefficients are: +1, -2, -5, +6.Count the sign changes: - From +1 to -2 (sign change)- From -2 to -5 (no change)- From -5 to +6 (sign change)There are 2 sign changes, indicating up to 2 positive real roots, possibly fewer (2 or 0 positive real roots).

Step 2: Identify the number of sign changes for negative roots

To find the possible number of negative roots, evaluate \(f(-x)\). Substitute \(-x\) into the original function:\[f(-x) = (-x)^3 - 2(-x)^2 - 5(-x) + 6 = -x^3 - 2x^2 + 5x + 6\]The coefficients are -1, -2, +5, +6. Count the sign changes:- From -1 to -2 (no change)- From -2 to +5 (sign change)- From +5 to +6 (no change)There is 1 sign change, indicating 1 possible negative real root.

Step 3: Graph the polynomial to verify the number of roots

Graph the polynomial \(f(x) = x^3 - 2x^2 - 5x + 6\) using a graphing tool or software.The graph shows the points where the curve crosses the x-axis. Check visually to identify the actual roots. The graph should clearly show that there is one positive root and one negative root.

Key Concepts

Polynomial Roots

Polynomial roots are the solutions to the polynomial equation set to zero. In simpler terms, they are the values of \(x\) where the polynomial \(f(x)\) crosses the x-axis. Understanding how to find these roots can reveal important features of the polynomial's graph.
To find these solutions, we set the polynomial \[f(x) = x^3 - 2x^2 - 5x + 6 = 0\]and solve for \(x\).
Roots can be found using various methods:

• Factoring: In some cases, you can factor the polynomial into simpler expressions to find the roots directly.
• Quadratic Formula: When dealing with second-degree polynomials, this formula can be applied.
• Numerical Methods: For higher-degree polynomials, numerical methods like the Newton-Raphson method might be required.

Beyond calculating, observing the graph of the polynomial is another way to find and confirm its roots.

Positive and Negative Roots

Determining the number of positive and negative roots is made simpler using Descartes' Rule of Signs. This rule is valuable because it provides possible counts of positive or negative roots without intricate calculations.
**Positive Roots**
To determine the number of positive roots, we examine the coefficients' sign changes in the polynomial \(f(x)\). For our polynomial, we noted sign changes in the series \(+1, -2, -5, +6\). There were two sign changes:
- From \(+1\) to \(-2\)
- From \(-5\) to \(+6\)
This suggests up to two or zero positive roots.
**Negative Roots**
To explore possible negative roots, substitute \(-x\) into the polynomial, resulting in new coefficients \(-1, -2, +5, +6\). The evaluation of these coefficients shows a single sign change:
- From \(-2\) to \(+5\)
This indicates there could be exactly one negative root.
By using Descartes' Rule of Signs, we pinpoint the potential number of positive and negative roots before graphing.

Graphing Polynomials

Graphing polynomials is a vital step to visually verify the actual number of roots suggested by Descartes' Rule of Signs. Creating a graph offers a clear picture of how the polynomial behaves along the x-axis.
When graphing \(f(x) = x^3 - 2x^2 - 5x + 6\), keep in mind these aspects:

• Intercepts: Look for where the curve crosses the x-axis. These points confirm the roots of the equation.
• End Behavior: Because this polynomial is cubic, it means that one of its ends goes to \(+∞\) and the other to \(-∞\).
• Turning Points: There could be up to \(n-1\) (where \(n\) is the degree of the polynomial) turning points that affect its shape and direction.

In this polynomial graph, you should identify one positive root and one negative root where the curve crosses the x-axis. Graphing provides a straightforward way to not only ascertain potential solutions but also confirm the actual quantity of roots present.