Problem 90 - Algebra and Trigonometry

Question

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ x^{5}+2.00 x^{4}+0.96 x^{3}+5.00 x^{2}+10.00 x+4.80=0 $$

Step-by-Step Solution

Answer

The real solutions are the x-values where the graph intercepts the x-axis, found using the device's trace/calculate function.

Step 1: Enter the Equation into the Graphing Device

Enter the given polynomial equation \(x^5 + 2.00x^4 + 0.96x^3 + 5.00x^2 + 10.00x + 4.80 = 0\) into your graphing calculator or software. Ensure all coefficients are entered correctly.

Step 2: Graph the Equation

Plot the equation on the graphing device. Set an appropriate viewing window, typically a range of \(x\) values from -10 to 10 and \(y\) values that allow a clear view of where the graph crosses the x-axis.

Step 3: Identify x-intercepts

Observe where the graph intersects the x-axis. These points represent the real solutions of the equation. Note that intersections could occur at multiple points or none in some cases.

Step 4: Use the Trace or Calculate Function

Utilize the "trace" or "calculate" function on your graphing device to find the accurate x-intercepts. Navigate to each point where the graph crosses the x-axis and use the device to find the x-values to two decimal places.

Key Concepts

Real Solutions

In mathematics, real solutions of a polynomial equation are the values of the variable that make the equation equal to zero. These are also known as roots or zeros of the polynomial. For real solutions:

• The equation has been satisfied without resorting to complex or imaginary numbers.
• They are the x-values at which the graph of the polynomial touches or crosses the x-axis.

Understanding real solutions is crucial because they often represent meaningful quantities in real-world problems. For the given problem, finding the real solutions involves knowing where the graph of the polynomial equation equals zero. Thus, focusing on these real points rather than imaginary ones provides clear insight into where real values solve the equation.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical concepts. For polynomial equations, it helps to:

• Input your equation correctly to represent it graphically.
• Adjust the viewing window to see important parts of the graph, such as where it crosses the x-axis.
• Use the tracing or calculating functions to precisely identify x-intercepts.

Many graphing calculators allow you to zoom and pan on the graph, making it easier to spot critical points like intercepts. They are indispensable for solving complex equations where the graphical representation provides insights that can be missed with manual calculations.

X-intercepts

The x-intercepts of a polynomial function are pivotal as they indicate where the graph intersects the x-axis. At these intercepts:

• The y-value of the function is zero.
• They correspond to the real solutions of the polynomial equation.
• In practical terms, they help find solutions that are applicable in real situations.

To locate x-intercepts accurately with a graphing calculator, it's essential to use functions like 'trace' or 'calculate,' which can provide precise x-values at the points of intersection. It’s these intercepts that are of utmost importance when identifying the real roots of a given polynomial.

Polynomial Functions

Polynomial functions represent expressions containing variables raised to various powers, along with coefficients. They are fundamental in mathematics, with varied applications:

• Allow the embodiment of complex relationships and patterns in a mathematically manageable form.
• Can describe real-life scenarios, like physical phenomena, financial projections, and more.
• Offer diverse solutions, including real and potential imaginary ones.

In the equation given, a fifth-degree polynomial is present, suggesting up to five possible x-intercepts, which our graphing calculator can help us identify. Understanding the structure of polynomials helps assess the function's behavior, predict its graphs' trajectory, and identify all solutions (real or otherwise) efficiently.