Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=3 x^{4}+2 x^{3}-8 x^{2}-10 x-1$$

Step-by-Step Solution

Answer

The polynomial has 1 positive and 1 negative real zero.

Step 1: State Descartes' Rule of Signs

Descartes' Rule of Signs states that the number of positive real roots of \( P(x) \) is either exactly equal to the number of sign changes between consecutive non-zero coefficients, or less than this by an even number. Similarly, the number of negative real roots of \( P(x) \) can be determined by applying Descartes' Rule to \( P(-x) \).

Step 2: Apply Descartes' Rule for Positive Roots

Examine the polynomial \( P(x) = 3x^4 + 2x^3 - 8x^2 - 10x - 1 \). The signs of the coefficients are: positive, positive, negative, negative, negative. Count the number of sign changes: the coefficients change sign from \(+\) to \(-\) twice, so there are 2 sign changes, indicating there could be 2 or 0 positive real roots.

Step 3: Apply Descartes' Rule for Negative Roots

Consider the polynomial \( P(-x) = 3(-x)^4 + 2(-x)^3 - 8(-x)^2 - 10(-x) - 1 \). This simplifies to \( 3x^4 - 2x^3 - 8x^2 + 10x - 1 \). The signs of these coefficients are: positive, negative, negative, positive, negative. Count the sign changes: \(+\) to \(-\), \(-\) to \(+\), and \(+\) to \(-\) gives 3 changes, so there could be 3 or 1 negative real roots.

Step 4: Graph the Polynomial

Graph \( P(x) = 3x^4 + 2x^3 - 8x^2 - 10x - 1 \) using a graphing calculator or software. Observe where the graph crosses the x-axis to identify the actual number of positive and negative real zeros.

Step 5: Determine the Actual Numbers of Real Zeros

From graphing \( P(x) \), observe that the graph of \( P(x) \) crosses the x-axis at only one point in the positive region and at one point in the negative region. Thus, there is 1 positive real zero and 1 negative real zero.

Key Concepts

Polynomial Roots

Polynomial roots are the solutions to the equation formed when a polynomial is set equal to zero. If you have a polynomial like \( P(x) = 3x^4 + 2x^3 - 8x^2 - 10x - 1 \), the roots of this polynomial are the values of \( x \) for which \( P(x) = 0 \). Roots can be real or complex numbers. In many cases, finding the roots of a polynomial can tell us a lot about its graph and behavior.

• Real roots are values that make the polynomial equal to zero and lie on the real number line.
• Complex roots occur in conjugate pairs and do not appear on the real number line.

To identify these roots effectively, tools like Descartes' Rule of Signs or graphing can be extremely helpful.

Real Zeros

Real zeros of a polynomial are the values of \( x \) where the graph intersects the x-axis. For instance, when investigating the polynomial \( P(x) = 3x^4 + 2x^3 - 8x^2 - 10x - 1 \), we are interested in identifying where \( P(x) = 0 \).

• Positive real zeros are the x-values where the polynomial crosses the x-axis on the positive side.
• Negative real zeros occur when the polynomial crosses on the negative side of the x-axis.

Using Descartes' Rule of Signs, we can predict the possible number of positive and negative real zeros. For positive zeros in our polynomial, with two sign changes in the sequence of coefficients, we might have 2 or 0 positive real zeros. For negative zeros, by substituting \( x \) with \( (-x) \), the rule suggests 3 or 1 negative real zeros. Graphing helps confirm these predictions.

Graphing Polynomials

Graphing polynomials is a vital step in determining the behavior and real roots of a polynomial equation. When graphing \( P(x) = 3x^4 + 2x^3 - 8x^2 - 10x - 1 \), visual inspection allows us to see where the graph crosses the x-axis.

• The points where the graph intersects the x-axis are the real zeros.
• Observing the graph, we can confirm the number of positive and negative real zeros predicted by Descartes' Rule of Signs.

In this case, graphing reveals that the polynomial has 1 positive real zero and 1 negative real zero. Using graphing software or a graphing calculator can simplify this process and provide a clear visualization of these intersections. This graphical method is a powerful tool in validating the theoretical predictions derived from numerical methods like Descartes' Rule of Signs.