Problem 81 - A Graphical Approach to College Algebra

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. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

Step-by-Step Solution

Answer

2 or 0 positive and 3, 1, or 0 negative real zeros possible.

Step 1: Identify Sign Changes for Positive Real Zeros

To determine the possible number of positive real zeros, observe the polynomial \(P(x) = x^{5}+3x^{4}-x^{3}+2x+3\). Count the number of times the signs of the coefficients change from one term to the next. Here, the sequence of coefficients is \(1, +3, -1, +2, +3\). The sign changes occur from \(+3\) to \(-1\) at the third term, and from \(-1\) to \(+2\) at the fourth term. Therefore, there are two sign changes, suggesting that there could be 2 or 0 positive real zeros.

Step 2: Substitute -x and Identify Sign Changes for Negative Real Zeros

For negative real zeros, substitute \(-x\) into the polynomial: \(P(-x) = (-x)^{5}+3(-x)^{4}-(-x)^{3}+2(-x)+3\), simplifying to \(-x^{5}+3x^{4}+x^{3}-2x+3\). Now observe the sequence of coefficients: \(-1, +3, +1, -2, +3\). The sign changes occur from \(-1\) to \(+3\), from \(+1\) to \(-2\), and from \(-2\) to \(+3\), yielding three sign changes, which suggests there could be 3, 1, or 0 negative real zeros.

Step 3: Verify with the Graph

Graph the polynomial \(P(x) = x^{5}+3x^{4}-x^{3}+2x+3\) to verify the number of positive and negative real zeros. From the graph, observe where the polynomial crosses the x-axis. Analyze the intercepts: these crossings represent actual real zeros.

Key Concepts

Polynomials

Let's start with polynomials to lay a strong foundation. Polynomials are mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients.
In the polynomial used in the exercise, the variable is "x". A polynomial is expressed in the form:

• \( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)

Here, each \( a_i \) represents a coefficient and \( x^i \) is a term, with \( n \) being the degree, which is the highest power of the variable.
In our polynomial \( P(x) = x^{5}+3x^{4}-x^{3}+2x+3 \), the degree is 5.

Real Zeros

Real zeros are the values of x that make the polynomial equal to zero. They are the x-values where the graph of the polynomial touches or crosses the x-axis.
Identifying these zeros helps in understanding the behavior of the polynomial. To determine the possible number of positive and negative real zeros, we use Descartes' Rule of Signs. This rule observes the sign changes between consecutive coefficients in the polynomial's terms to predict the number.
Positive real zeros are identified by observing the original polynomial's coefficients' sign changes. For the polynomial \( P(x) = x^5 + 3x^4 - x^3 + 2x + 3 \), we see 2 sign changes, suggesting 2 or 0 positive real zeros are possible. For negative real zeros, substitute \(-x\) into the polynomial and simplify, checking for the same sign changes.
In this example, \( P(-x) = -x^5 + 3x^4 + x^3 - 2x + 3 \), with 3 sign changes indicating 3, 1, or 0 negative real zeros.

Graphing Polynomials

Graphing polynomials is a visual way to confirm our solutions and understand the roots or zeros of the polynomial. A graph provides an intuitive representation of where a polynomial crosses the x-axis. For a polynomial \( P(x) = x^{5}+3x^{4}-x^{3}+2x+3 \), you can plot it using graphing software or technology like a graphing calculator. Once graphed, the intersections with the x-axis help affirm the predicted number of real zeros: both positive and negative. When graphing, look for:

• Intercepts: These are where the graph crosses the axis and indicate real zeros.
• Turning Points: Points where the graph changes direction, relevant in polynomials of degree 5 used in this exercise.

By observing these features, we can see that the graph provides practical confirmation of real zeros, adding a visual dimension to our algebraic work.