Problem 5 - Precalculus

Question

You are given a polynomial equation \(f(x)=0 .\) According to the fundamental theorem of algebra each of these equations has at least one root. However, the fundamental theorem does not tell you whether the equation has any real-number roots. Use a graph to determine whether the equation has at least one real root. Note: You are not being asked to solve the equation. $$x^{3}-3 x^{2}+3=0$$

Step-by-Step Solution

Answer

The graph shows the polynomial intersects the x-axis, indicating at least one real root.

Step 1: Understand the Polynomial

The given polynomial is \( f(x) = x^3 - 3x^2 + 3 \). This is a cubic equation, which means it is of degree 3. According to the fundamental theorem of algebra, a polynomial of degree 3 has exactly 3 roots, although some may be complex.

Step 2: Consider the Real-Number Root Possibility

The equation might have real, complex, or a combination of both types of roots. To determine if there are any real roots, we can graph the polynomial and observe where it intersects the x-axis.

Step 3: Graph the Polynomial

Plot the graph of the polynomial function \( f(x) = x^3 - 3x^2 + 3 \). Use a graphing calculator or software to accurately plot the equation. Look for points where the graph intersects the x-axis.

Step 4: Analyze the Graph

Upon graphing, observe the behavior of the curve. If the curve crosses the x-axis at any point, the x-coordinate of those points are real roots of the equation. In this example, the graph shows that it crosses the x-axis around \( x \approx 0.8 \), indicating the presence of a real root.

Key Concepts

Understanding Polynomial Equations

Polynomial equations, like the one in this exercise, involve algebraic expressions made up of variables and coefficients. These variables are often raised to whole number powers. The equation provided, \( f(x) = x^3 - 3x^2 + 3 \), is a straightforward example.

A polynomial equation is defined by its degree, which is the highest power of the variable present in the expression. In this case, the degree is 3, indicating that it is a cubic polynomial.

Key points about polynomial equations include:

• They can have multiple terms, like constants, linear terms (e.g., \( x \)), quadratic terms (e.g., \( x^2 \)), and so on.
• The degree of the polynomial tells us the maximum number of roots the equation can have.
• According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots, though not all need be real numbers.

Understanding these basics helps in analyzing the behavior of polynomial equations and in predicting the nature of their roots.

Exploring Cubic Functions

Cubic functions like the one in our exercise, expressed as \( x^3 - 3x^2 + 3 = 0 \), showcase interesting properties in polynomial analysis. By definition, cubic equations involve variables raised to the third power. Here's what makes them unique:

Characteristics of cubic functions include:

• They are nonlinear equations and exhibit more complex behavior compared to linear and quadratic equations.
• Their graphs typically present one or two turning points, depending on their coefficients.
• They exhibit symmetry or a lack thereof, which can be helpful in visualizing their graphs.

To determine properties like intercepts and endpoints, graphing can be particularly useful. When graphing a cubic function:

• Observe the number of times the curve crosses the x-axis (real roots).
• Check how the graph behaves at extreme values of \( x \) to understand end-behavior.
• Look for changes in direction to identify turning points.

Graphing unveils much of the complexity within cubic functions, aiding the identification of real and complex roots.

Determining Real Roots

Real roots of polynomial equations are the solutions, or x-values, where the graph of the polynomial intersects the x-axis. For the equation \( x^3 - 3x^2 + 3 = 0 \), identifying real roots is straight forward through this approach:

To discover if a polynomial has real roots:

• Graph the equation using graphing software or a calculator.
• Note where the curve meets the x-axis, as these represent real roots of the equation.
• Real roots are identifiable, whereas complex roots do not cause crossings of the x-axis.

When you plot the function \( x^3 - 3x^2 + 3 \), you might see the curve intersect the x-axis at a point like \( x \approx 0.8 \).

The ability to determine the specific point where this occurs signifies that a real root exists.
Graph analysis can clarify the presence of real roots within polynomial equations, providing visual insight beyond algebraic manipulations.