Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=x^{4}+2 x^{3}-12 x^{2}+14 x-5 $$

Step-by-Step Solution

Answer

There are possibly 3, 1, or 0 positive roots and 1 negative root.

Step 1: Understanding Descartes' Rule of Signs

Descartes' Rule of Signs helps us determine the possible number of positive and negative real roots of a polynomial. For positive roots, we look at the number of sign changes in the coefficients of the polynomial as written. For negative roots, we substitute x = -x into the polynomial and then count the sign changes.

Step 2: Count Sign Changes for Positive Roots

Examine the coefficients of the polynomial, which are [1, 2, -12, 14, -5]. Count the sign changes between consecutive terms: - From 1 to 2 (no change) - From 2 to -12 (1 change) - From -12 to 14 (1 change) - From 14 to -5 (1 change). This results in 3 sign changes, indicating there might be 3, 1, or no (0) positive roots.

Step 3: Substitute x with -x for Negative Roots

Substitute -x into f(x): \[ f(-x) = (-x)^4 + 2(-x)^3 -12(-x)^2 + 14(-x) - 5 \] simplifies to \[ f(-x) = x^4 - 2x^3 - 12x^2 - 14x - 5 \] The coefficients are now [1, -2, -12, -14, -5]. Count the sign changes:

Step 4: Count Sign Changes for Negative Roots

Examine the coefficients of the modified polynomial, which are [1, -2, -12, -14, -5]. Count the sign changes: - From 1 to -2 (1 change) - From -2 to -12 (no change) - From -12 to -14 (no change) - From -14 to -5 (no change). This results in 1 sign change, indicating there might be 1 negative root.

Step 5: Confirm with Graph

By graphing the function, you can visually confirm the number of real positive and negative roots. You might observe that the function crosses the x-axis three times if there are 3 positive roots, one positive root if it crosses once, and one negative root from the substitution analysis.

Key Concepts

Polynomial Roots

Polynomial roots are the values of the variable (in this case, \(x\)) that make the polynomial equal to zero. They are essentially the solutions of the polynomial equation. In our exercise, we have the polynomial \(f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5\). The goal is to find the real roots of this polynomial, which can either be positive, negative, or complex (not real).
Using Descartes' Rule of Signs is an efficient way to estimate the number of positive and negative real roots without solving the equation directly. This provides a strategic starting point for a more detailed analysis of the polynomial.

Sign Changes in Coefficients

The concept of sign changes refers to how the sign of the coefficients change as you move from one term to the next in a polynomial. For example, moving from a positive to a negative coefficient constitutes a sign change.
In our polynomial \([1, 2, -12, 14, -5]\), Descartes' Rule of Signs suggests that the potential number of positive real roots corresponds to these sign changes. Calculating these gives us three sign changes:

• From \(1\) to \(2\): no change
• From \(2\) to \(-12\): one change
• From \(-12\) to \(14\): one change
• From \(14\) to \(-5\): one change

This analysis informs us that there might be either 3, 1, or 0 positive real roots. The variations in possible roots occur because complex roots, which always come in conjugate pairs, may cancel out some positive roots.

Positive and Negative Roots

To determine negative roots, we replace \(x\) with \(-x\) in the polynomial and analyze the resulting sign changes. This substitution reflects the behavior of the polynomial for negative values of \(x\).
For our example, substituting gives us the polynomial \(x^4 - 2x^3 - 12x^2 - 14x - 5\). The coefficients are \([1, -2, -12, -14, -5]\), allowing us to count sign changes:

• From \(1\) to \(-2\): one change
• From \(-2\) to \(-12\): no change
• From \(-12\) to \(-14\): no change
• From \(-14\) to \(-5\): no change

This indicates the possibility of only one negative real root.
Always remember, when counting sign changes after substitution, each sign change translates to a potential root, with the same rules about imaginary roots impacting possible numbers.

Graphing Functions

Graphing the polynomial function provides a visual confirmation of our analysis on the number of real roots. By plotting \(f(x)\), you observe where the curve crosses the x-axis, which represents the roots.
In the graph for \(f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5\), each intersection with the x-axis corresponds to a real root. If the graph crosses the x-axis at three points in the positive region, it indicates three positive real roots. If it only crosses once, there may be one positive root. Likewise, a single crossing in the negative region supports the finding of one negative root.

• Graphing helps to visually confirm theoretical predictions.
• Discrepancies can prompt a reevaluation, accounting for complex roots.
• Graph interpretation remains crucial for comprehensive understanding.

Graphing is a powerful tool to ensure our calculations, while rooted in theory, correspond to observable outcomes.