Problem 23 - Applied Calculus

Question

23-24. Solve each equation using a graphing calculator. Round answers to two decimal places. $$ x^{5}-x^{4}-5 x^{3}=0 $$

Step-by-Step Solution

Answer

The roots of the equation are \( x = 0 \), \( x \approx 2.79 \), and \( x \approx -1.79 \).

Step 1: Recognize the equation

Identify the given polynomial equation: \( x^5 - x^4 - 5x^3 = 0 \). This is a fifth-degree equation, suggesting that it might have up to five real roots.

Step 2: Factor out common terms

Notice that each term in the equation shares an \( x^3 \) factor. Factor \( x^3 \) from the entire equation: \( x^3(x^2 - x - 5) = 0 \). This indicates a trivial root at \( x = 0 \), and the remaining factor to solve is \( x^2 - x - 5 = 0 \).

Step 3: Use graphing calculator for quadratic

To solve \( x^2 - x - 5 = 0 \), graph the function \( y = x^2 - x - 5 \) using a graphing calculator. Look for points where the graph intersects the x-axis, as these points are the real roots of the quadratic equation.

Step 4: Identify real roots on the graph

Examine the graph and find the points where it crosses the x-axis. These points, approximately located, give the non-zero solutions for the equation. Calculate and round these to two decimal places using the calculator's trace or root finding functions.

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler "factors" that can be multiplied together to achieve the original polynomial. This process is much like factoring numbers, but here we're working with variables and coefficients. In the given polynomial equation, \(x^5 - x^4 - 5x^3 = 0\), we noticed that each term shares the factor \(x^3\).
This common factor can be factored out, simplifying our equation to \(x^3(x^2 - x - 5) = 0\).
When we successfully extract common factors from the polynomial, one of the solutions, in this case, becomes apparent: \(x = 0\).
Factoring is a crucial step in solving polynomial equations as it reduces the complexity and can reveal easier solutions or roots. Another simple factor that might emerge is a quadratic like \(x^2 - x - 5\), which can be solved separately to find additional roots.

• Determine the common factor across all terms.
• Factor it out to simplify the equation.
• Solve each factor individually to find all potential solutions.

Graphing Calculator Use

Using a graphing calculator can be a powerful tool in solving polynomial equations, especially when factors aren't easily solved by hand. These devices help visualize and pinpoint solutions that lie on real-number lines. When dealing with the equation \(x^2 - x - 5 = 0\), graphing the function \(y = x^2 - x - 5\) shows us exactly where the graph intersects the x-axis. This intersection represents the real roots of the equation.
To find these intersections, you can utilize the graphing calculator's functions like tracing or finding roots, which usually allows you to zoom into the section of the graph where the roots appear.
Here are some steps to take when using a graphing calculator:

• Enter the function into the calculator.
• View the graph and identify where the curve crosses the x-axis.
• Use the calculator’s "zero" or "root" feature to precisely determine the x-intercepts.
• Round these solutions to two decimal places as needed.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They have the standard form \(ax^2 + bx + c = 0\). In our case, after factoring the original polynomial equation, the leftover problem is \(x^2 - x - 5 = 0\), which is a quadratic equation.
The real solutions of a quadratic can be found using different methods such as factoring, completing the square, using the quadratic formula, or graphically with a graphing calculator. Here, we already tried graphing to find the roots.
Let's quickly revisit how you could solve it analytically with the quadratic formula:
The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). For our equation \(a = 1\), \(b = -1\), and \(c = -5\).
Plugging these into the formula:
\[x = \frac{{1 \pm \sqrt{{(-1)^2 - 4(1)(-5)}}}}{2(1)}\] This gives us two potential solutions for \(x\), which can be accurately verified by checking these against the graphing tool results and rounding them to two decimal places.
Each of these approaches offers a different way to understand and solve quadratic equations efficiently.