Problem 47 - 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. \(f(x)=x^{4}-x^{2}-1\)

Step-by-Step Solution

Answer

1 or 0 positive roots; 1 or 0 negative roots; graph confirms 0 positive, 2 negative roots.

Step 1: Identify Sign Changes for Positive Roots

To determine the possible number of positive real roots using Descartes’ Rule of Signs, examine the signs of the coefficients of the polynomial function. The given polynomial function is \[ f(x) = x^4 - x^2 - 1. \]Here are the signs of its terms in order: Positive (for \(x^4\)), Negative (for \(-x^2\)), and Negative (for \(-1\)). We have one sign change from Positive to Negative. This indicates there is 1 or 0 positive real root.

Step 2: Identify Sign Changes for Negative Roots

Next, find the possible number of negative real roots by considering the polynomial \( f(-x) \). Substitute \(-x\) into the function:\[f(-x) = (-x)^4 - (-x)^2 - 1 = x^4 - x^2 - 1.\] The polynomial in terms of \(-x\) has the same signs as the original: Positive (for \((-x)^4\)), Negative (for \((-x)^2\)), and Negative (for \(-1\)). There is one sign change, indicating 1 or 0 negative real roots.

Step 3: Graph the Function

To confirm the number of positive and negative real roots, graph the polynomial \(f(x) = x^4 - x^2 - 1\). Look for the x-intercepts on the graph. When graphed, the x-intercepts indicate the actual roots of the function. By analyzing the graph, we notice it does not cross the x-axis at any positive values, confirming there are no positive real roots. The graph crosses the x-axis twice at two symmetrical points, indicating two negative real roots.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to whole number exponents and coefficients. In simpler terms, a polynomial is a sum of several terms, where each term is a product of a constant and a variable raised to a non-negative integer exponent. The polynomial \[f(x) = x^4 - x^2 - 1\] consists of three terms: \(x^4\), \(-x^2\), and \(-1\). This specific polynomial is a quartic polynomial, meaning the highest exponent of the variable, or degree, is four. Understanding the degree of a polynomial is crucial because it can tell us various things about the polynomial, such as its general shape or the maximum number of roots it can have. A polynomial of degree \(n\) can have up to \(n\) real roots. However, not all roots might be real, and this is where techniques like Descartes' Rule of Signs come in handy to help determine the number of positive and negative real roots possible.

Real Roots

Real roots of a polynomial are the values of \(x\) where the polynomial equals zero. For instance, if \(f(x) = x^4 - x^2 - 1\), then finding the real roots involves figuring out for which values of \(x\) the expression equates to zero, meaning \(f(x) = 0\). We apply Descartes' Rule of Signs to count the potential positive and negative real roots, but it doesn’t guarantee their existence or their exact values. - For this particular polynomial, as discussed in the exercise, there are no positive real roots.- The graph provides visual confirmation and shows that the function crosses the x-axis twice, both at negative values of \(x\). These are the negative real roots of the polynomial. This illustrates that, while Descartes' Rule can propose possible root counts, only graphing or further algebraic methods can definitively confirm where these roots actually lie on the number line.

Sign Changes

Sign changes in a polynomial function refer to the transitions from plus to minus (or vice versa) among the terms arranged by increasing or decreasing power of the variable. These changes are crucial due to Descartes' Rule of Signs, which relates them to the number of roots.To count sign changes:1. Write down the polynomial with terms in standard order for positive roots.2. Check for the signs of each term's coefficient, noting where they switch from positive to negative or negative to positive.3. This switch is a 'sign change'.In the polynomial \[f(x) = x^4 - x^2 - 1\], there is one sign change from the term with a positive coefficient \(x^4\) to the next term \(-x^2\) with a negative coefficient, identifying one or zero possible positive roots.For negative roots, substitute \(-x\) into the polynomial and repeat the process. Interestingly, our original exercise found no sign change difference for \(f(-x)\), indicating one or zero negative real roots initially, yet the graph confirmed two. This unusual result underscores how crucial context or visualization might sometimes validate or reveal new insights beyond theoretical speculation.