Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, use Descartes' Rule to determine the possible number of positive and negative solutions. Confirm with the given graph. $$ f(x)=x^{3}-1 $$

Step-by-Step Solution

Answer

One positive root, no negative roots; confirmed by graphing.

Step 1: Identify possible number of positive real roots

The given polynomial is \( f(x) = x^3 - 1 \). To find the positive real roots, we will count the number of times the coefficients of the polynomial change sign. After expanding, the terms are \( x^3, 0x^2, 0x, -1 \). Here, from \( x^3 \) to \( -1 \), there is one sign change. Therefore, there is one possible positive real root according to Descartes' Rule.

Step 2: Identify possible number of negative real roots

To determine possible negative real roots, evaluate \( f(-x) \). This gives \( f(-x) = (-x)^3 - 1 = -x^3 - 1 \). The terms are \( -x^3, 0x^2, 0x, -1 \). Here, there is no sign change (both terms are negative), so there are no possible negative real roots.

Step 3: Confirming roots with a graph

Graph the function \( f(x) = x^3 - 1 \) to confirm roots. The graph crosses the x-axis at \( x = 1 \), confirming the existence of exactly one positive real root. Since the graph does not cross or touch the x-axis at any other points, there are no other real roots.

Key Concepts

Polynomial Roots

Polynomial roots are the values of the variable that make the polynomial equal to zero. In simple terms, they are the solutions to the equation formed by setting the polynomial equal to zero. For example, if we have a polynomial function \( f(x) = x^3 - 1 \), finding its roots means solving the equation \( x^3 - 1 = 0 \). This equation can be solved by factoring, graphing, or using analytical methods.
This results in the roots: \( x = 1 \). In this specific case, there is only one real root.
Roots can be real or complex, and real roots can either be positive or negative. Each polynomial can have as many roots as its degree, which is the highest power of the variable in the polynomial. However, not all of these roots have to be real; some might be complex numbers.

Positive Real Roots

In the exploration of polynomial roots, it's important to identify if and how many of these roots are positive real numbers. Positive real roots are those roots that are greater than zero. Descartes' Rule of Signs is a useful tool to determine the possible number of positive real roots a polynomial may have.
For a given polynomial, like \( f(x) = x^3 - 1 \), you examine the polynomial's terms and observe the sign changes between consecutive terms.

• Each sign change indicates a potential positive real root.
• Count the transitions from a positive coefficient to a negative one or vice versa.

For our function, \( f(x) = x^3, 0, 0, -1 \), there is only one sign change (from \( x^3 \) to \( -1 \)).
Thus, there is exactly one positive real root. This is verified by the graph, which shows the curve crossing the x-axis at \( x = 1 \). After checking, there are no other positive intercepts on the x-axis.

Negative Real Roots

Negative real roots are values of the variable that are less than zero and make the polynomial equal to zero. To determine the possible number of negative real roots for a polynomial, Descartes' Rule of Signs can be applied by evaluating \( f(-x) \). By substituting \( -x \) into the given function, you change each term and examine the resulting polynomial.
It shows how the signs of each term might be affected:

• For the function \( f(x) = x^3 - 1 \), substitute \(-x\) to get \( f(-x) = -x^3 - 1 \).
• Observe the sequence of signs \( -x^3, 0, 0, -1 \).

Notice that there are no sign changes.
Hence, according to Descartes' Rule, there are no negative real roots.
The absence of a negative real root suggests that the graph of the polynomial will not intersect the x-axis for any negative value of \( x \), which can be confirmed visually by checking its graph.

Graphing Functions

Graphing functions is a practical method to validate the results about roots obtained through algebraic means. A graph provides a visual representation of a polynomial and its roots. This helps to confirm whether there are any points where the graph intersects the x-axis, indicating real roots.
For the polynomial \( f(x) = x^3 - 1 \), when plotted, it shows a curve that intersects the x-axis exactly once at \( x = 1 \), aligning with our earlier findings for positive real roots.
To graph a polynomial, follow these steps:

• Plot enough points by substituting various \( x \) values into the polynomial to understand its behavior.
• Look for intercepts—where the graph crosses the axes.
• Check the end behavior, which is determined by the leading term—in this case, \( x^3 \), leading it to rise and fall steeply.

Graphing provides insights into the function's nature and clarifies the findings from Descartes' Rule and other algebraic tests. This combination of algebraic reasoning and visual verification ensures a comprehensive understanding of polynomial roots.