Problem 101 - Precalculus with Limits

Question

In Exercises 99-106, use the formula on page 474 to find all the solutions of the equation and represent the solutions graphically. \(x^5 + 243 = 0\)

Step-by-Step Solution

Answer

The solution to the equation \(x^5 + 243 = 0\) is \(x = -3\). This result is clearly visible on a graph of the function, where the function crosses the x-axis at \(x = -3\).

Step 1: Rearrange the Equation

First, rearrange the equation to isolate \(x^5\) on one side. This can be done by subtracting 243 from both sides of the equation, yielding \(x^5 = -243\).

Step 2: Take the fifth root of both sides

Since \(x\) is raised to the power of 5, take the fifth root of both sides to solve for \(x\). The fifth root of -243 is -3 (since \(-3^5 = -243\)), so \(x = -3\).

Step 3: Represent the solutions graphically

The solution to the equation can be shown on a graph. Plot a point at \(x = -3\) on the x-axis. This is the solution to the equation. Since the exponent is odd, the graph will cross the x-axis at this point, and the end behavior of the function will be in opposite directions.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions involving variables raised to natural number exponents, combined using operations such as addition, subtraction, and multiplication. The general form of a polynomial equation is expressed as: \[ ax^n + bx^{n-1} + cx^{n-2} + \ldots + k = 0 \]where each term consists of a coefficient and a variable raised to an exponent. In the given exercise, the polynomial is \(x^5 + 243 = 0\). This is a fifth-degree polynomial because the highest exponent of the variable \(x\) is 5.

• The degree of the polynomial determines the number of roots or solutions, which, for a real polynomial, could be real or complex.
• Each term in the polynomial represents a part of the equation that can contribute to the shape and position of its graph.
• Solving a polynomial involves finding values of \(x\) that satisfy the equation. In this exercise, we solve by isolating \(x^5\), giving us \(x^5 = -243\).

Understanding polynomial equations lays the foundation for solving complex equations and interpreting their behavior when graphed.

Graphical Representation

Graphical representation helps to visually interpret the behavior of a polynomial equation. When plotting the equation \(x^5 + 243 = 0\), it translates into observing the curve where the equation equals to zero.

• A graph of a polynomial gives insight into the roots and nature of the equation. The x-intercepts of the graph correspond to the roots of the equation.
• The shape of the graph can vary greatly with the degree of the polynomial. Higher-degree polynomials can have more bends and twists, corresponding to their complex roots.
• For our equation, after taking the fifth root, we place a point on the graph at \(x = -3\). This point is where the graph crosses the x-axis.

Graphing helps in not only finding the roots but also understanding the overall behavior and end patterns of the polynomial function. For a fifth-degree polynomial like \(x^5 + 243 = 0\), the graph will display an end behavior where one end extends towards positive infinity and the other towards negative infinity, reflecting its odd-degree nature.

Roots of Equations

The roots of an equation are the solutions where the equation equals zero. These are the values of \(x\) for which \(f(x) = 0\). In our equation, we found that the fifth root of \(-243\) is \(-3\), so \(x = -3\) is the only real root.

• Roots are crucial as they provide the values that balance the equation.
• In polynomial equations, the number of roots (counting multiplicities and considering complex roots) is equal to the degree of the polynomial.
• The root \(-3\) tells us at this point, the polynomial equation is zero. In graphical terms, this is where the graph intersects the x-axis.

Understanding the roots includes both finding them algebraically and interpreting them graphically. For polynomial equations like \(x^5 + 243 = 0\), solving for roots is pivotal for fully comprehending the function's behavior and making accurate predictions about its graph's trajectory.