Problem 78 - 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. $$P(x)=x^{3}+2 x^{2}+x-10$$

Step-by-Step Solution

Answer

1 positive real zero, 0 negative real zeros.

Step 1: Identify Sign Changes for Positive Real Zeros

Using Descartes' Rule of Signs, we look at the polynomial \(P(x) = x^3 + 2x^2 + x - 10\). We need to count the sign changes in the coefficients of the terms. The signs of the coefficients are: \(+\) for \(x^3\), \(+\) for \(2x^2\), \(+\) for \(x\), and \(-\) for \(-10\). This yields one sign change from \(+\) to \(-\), indicating that there could be either 1 positive real zero.

Step 2: Identify Sign Changes for Negative Real Zeros

To determine the possible numbers of negative real zeros, replace \(x\) with \(-x\) and simplify \(P(-x)\). This gives us \((-x)^3 + 2(-x)^2 + (-x) - 10 = -x^3 + 2x^2 - x - 10\). For \(P(-x)\), the signs of the coefficients change as follows: \(-\), \(+\), \(-\), \(-\). We observe two sign changes: from \(-\) to \(+\) and then from \(+\) to \(-\), indicating there could be 0 or 2 negative real zeros.

Step 3: Graph the Function to Identify Actual Zeros

Graph the function \(P(x) = x^3 + 2x^2 + x - 10\) using a graphing calculator or software. Observe where the graph intersects the x-axis. The intersections signify real zeros of the function.

Step 4: Analyze the Graph

Upon analyzing the graph, we determine the actual number of real zeros. The graph intersects the x-axis at one point indicating one positive real zero. It does not cross back over the x-axis in the negative region, suggesting there are no negative real zeros.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the values of \(x\) where the polynomial equals zero. In simpler terms, they are the points where the graph of the function intersects the x-axis. For a polynomial, these zeros might be positive, negative, or even complex numbers which do not touch the x-axis at all.

To find the possible real zeros, we often use Descartes' Rule of Signs, a handy tool that examines the sign changes in the polynomial’s terms to give us the potential count of positive and negative real zeros.

In the polynomial \(P(x) = x^3 + 2x^2 + x - 10\), there is a pattern of sign changes from \(x^3\) to constant \(-10\). This change from positive to negative shows that there can be 1 positive real zero. By evaluating \(-x\) in place of \(x\), we check for negative real zeros. In our example, this results in two possible negative real zeros due to the two sign changes \(- \rightarrow + \rightarrow -\). However, these are only possibilities; actual zeros need further verification.

Polynomial Functions

Polynomial functions are expressions involving a sum of powers in one or more variables multiplied by coefficients. They are fundamental because they represent many real-world situations and mathematical concepts. The general form for a polynomial in one variable, like \(P(x)\), is written as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0\), where \(a_n\) represents coefficients, and \(n\) is a non-negative integer.

The degree of the polynomial, given by the highest power \(n\), informs us about the shape of the curve and the maximum number of real zeros the polynomial could have. For \(P(x) = x^3 + 2x^2 + x - 10\), the degree is 3, indicating it can have up to three real zeros. However, not all of these zeros must be real or distinct.

This understanding of polynomials allows us to use various techniques, like Descartes' Rule of Signs and graphing, to deduce information about the roots of the function and better understand their graphical behavior.

Graphing Functions

Graphing functions is a powerful way to visualize the solutions to equations and understand the behavior of the function over a range of values. For a polynomial like \(P(x) = x^3 + 2x^2 + x - 10\), using a graphing tool can quickly reveal the nature and position of its real zeros.

When graphed, this polynomial will display a curve that may intersect the x-axis at points, which visually represent the real zeros we’ve been calculating. By plotting \(P(x)\), we can observe one intersection on the positive side, confirming a single positive real zero. Meanwhile, no intersections appear on the negative side, supporting a conclusion of zero negative real zeros.

Graphical analysis is useful because it provides immediate observation that complements numerical and algebraic methods. Being able to see where a graph crosses the x-axis helps reinforce findings from Descartes' Rule of Signs, offering a complete perspective on the function's behavior.