Problem 50 - 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. \(f(x)=x^{4}+2 x^{3}-12 x^{2}+14 x-5\)

Step-by-Step Solution

Answer

1 positive and 1 negative real root, confirmed by graphing.

Step 1: Identify Changes in Sign for Positive Roots

To find the number of possible positive real roots, apply Descartes' Rule of Signs by examining the changes in the sign of the coefficients of the polynomial \(f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5\). The signs are +1 (for \(x^4\)), +2 (for \(x^3\)), -12 (for \(x^2\)), +14 (for \(x\)), and -5 (constant term). The sign changes are: `+` to `-`, `-` to `+`, and `+` to `-`, totaling 3 sign changes. Therefore, there could be 3 or 1 positive real roots.

Step 2: Identify Changes in Sign for Negative Roots

To find the number of possible negative real roots, evaluate the polynomial at \(f(-x) = (-x)^4 + 2(-x)^3 - 12(-x)^2 + 14(-x) - 5 = x^4 - 2x^3 - 12x^2 - 14x - 5\). Here, the signs of coefficients are: +1 (for \(x^4\)), -2 (for \(x^3\)), -12 (for \(x^2\)), -14 (for \(x\)), and -5. There is only 1 change in sign, from \(+1\) to \(-2\). Thus, there can be 1 negative real root.

Step 3: Determine Possible Combinations

From the analysis, we have found that there could be either 3 or 1 positive roots and there is exactly 1 negative root. Possible combinations for the number of real roots could be: - 3 positive and 1 negative (total 4 real roots) - 1 positive and 1 negative (with 2 complex roots making up the remaining solutions since the polynomial is degree 4).

Step 4: Graphical Confirmation

Graph the polynomial \(f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5\). By observing the graph, identify the points where the curve intersects the x-axis. Each intersection represents a real root. You will find that the graph confirms exactly 2 real roots (1 positive, 1 negative) and thus 2 complex roots.

Key Concepts

Positive Real Roots

To find the number of possible positive real roots of a polynomial, we utilize Descartes' Rule of Signs. This rule helps us predict the number of positive real roots in a polynomial equation by examining the sign changes in its coefficients. In the polynomial function given, \( f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5 \), we list the coefficients and observe the following sign changes:

• From \(+1\) (coefficient of \(x^4\)) to \(+2\) (coefficient of \(x^3\)), no change;
• From \(+2\) to \(-12\) (coefficient of \(x^2\)), a change occurs;
• From \(-12\) to \(+14\) (coefficient of \(x\)), a change occurs;
• From \(+14\) to \(-5\) (constant term), a change occurs.

Here, we identify 3 changes in sign, indicating that there could be 3 or 1 positive real root(s).

The discrepancy in possible roots arises because real roots count can also include complex roots paired with their conjugates, thanks to the Fundamental Theorem of Algebra. Thus, for a polynomial of degree 4, we initially suspect up to 3 positive roots.

Negative Real Roots

For determining the number of negative real roots, Descartes' Rule of Signs requires us to substitute \(-x\) into the original polynomial. This gives us a transformed polynomial, \(f(-x) = (-x)^4 + 2(-x)^3 - 12(-x)^2 + 14(-x) - 5\), simplifying to:
\[ f(-x) = x^4 - 2x^3 - 12x^2 - 14x - 5 \]
Evaluating this polynomial's coefficients, we observe:

• From \(+1\) (coefficient of \(x^4\)) to \(-2\) (coefficient of \(x^3\)), there is a change;
• The remaining coefficients are negative, leading to no further changes in sign.

This indicates 1 sign change, suggesting that there is exactly 1 negative real root.

This result aligns with the fact that all expected roots (real and complex) add up to the degree of the polynomial, which in this case is 4. Thus, beyond the 1 negative root, the remaining solutions must consist of the other roots (either positive or complex).

Graphical Method

The graphical method offers a visual confirmation of the polynomial's real root structure predicted by Descartes' Rule of Signs. To achieve this, plot the polynomial \( f(x) = x^4 + 2x^3 - 12x^2 + 14x - 5 \) on a coordinate plane and observe its intersecting points with the x-axis.

When a polynomial touches or crosses the x-axis in a graph, these points are where real roots exist. Checking the intersections:

• Look for the curve crossing the x-axis from above or below; each crossing corresponds to a real root.
• Count the number of crossings: in this exercise, the graph shows two intersections at the x-axis—one representing a positive real root and one a negative real root.

This graphical evidence confirms 1 positive and 1 negative real root, accompanied by 2 non-real complex roots to satisfy the polynomial's degree. This visual representation often provides a refreshing perspective for learning, complementing the algebraic approach of root determination.