Problem 40 - Precalculus Mathematics for Calculus

Question

Solve the equation both algebraically and graphically. $$2 x^{5}-243=0$$

Step-by-Step Solution

Answer

The solution is approximately \( x = 2.9 \).

Step 1: Rearrange the Equation

Begin by rearranging the given equation to isolate the polynomial term on one side:\[ 2x^5 = 243 \]. To simplify further, divide both sides by 2 to get:\[ x^5 = 121.5 \].

Step 2: Solve Algebraically Using Fifth Roots

To solve algebraically, you need to find \( x \) such that \( x^5 = 121.5 \). Take the fifth root of both sides:\[ x = \sqrt[5]{121.5} \].Using a calculator, compute the fifth root:\[ x \approx 2.9 \].

Step 3: Graphical Solution Setup

To solve the equation graphically, represent it as a function \( f(x) = 2x^5 - 243 \). You want to find the \( x \)-values where \( f(x) = 0 \).

Step 4: Plot the Graph

Using graphing software or a calculator, plot \( y = 2x^5 - 243 \) over an appropriate range to observe where the graph crosses the \( x \)-axis. This intersection is the solution.

Step 5: Identify Intersection Points

Upon plotting, observe that the graph intersects the \( x \)-axis at the point \( x \approx 2.9 \), confirming the algebraic result.

Key Concepts

Algebraic Solutions

Algebraic solutions involve manipulating an equation to directly solve for the variable. Let's break down the steps taken to solve the polynomial equation \(2x^5 - 243 = 0\) using algebraic methods. First, you need to rearrange the equation into a more simplified form. This involves isolating the polynomial term, \(2x^5\), on one side of the equation. You then have:

• \(2x^5 = 243\)

To solve for \(x\), you simplify further by dividing by 2:

• \(x^5 = 121.5\)

The key to solving this equation algebraically is finding the fifth root of both sides, which gets us:

• \(x = \sqrt[5]{121.5}\)

Calculating this results in \(x \approx 2.9\).Remember, the fifth root is essential here because we have a fifth-degree polynomial equation. By isolating \(x\), we're able to find the precise value that makes the equation true.

Graphical Solutions

To solve an equation graphically, you need to visualize its function on a graph. For our equation \(2x^5 - 243=0\), the corresponding function is \(f(x) = 2x^5 - 243\). The goal is to find where the graph of this function crosses the \(x\)-axis. These crossing points are the solutions for \(x\). First, represent the problem as a function:

• \(f(x) = 2x^5 - 243\)

Next, using graphing software or a calculator, plot this function. Ensure that you set a suitable range that includes possible solution points. The point where the graph intersects the \(x\)-axis indicates the solution to the equation. In this case, after plotting, observe the graph crosses the \(x\)-axis at approximately \(x = 2.9\). This graphical intersection confirms the algebraic solution, showing a consistent result across different solving methods.

Fifth Roots

Fifth roots are crucial in solving higher-degree polynomial equations, like our equation \(x^5 = 121.5\). The fifth root of a number is the value that, when multiplied by itself five times, gives the original number. When taking the fifth root of both sides of an equation, like \(x^5 = 121.5\), you're simplifying it to:

• \(x = \sqrt[5]{121.5}\)

This operation is important in finding the specific value of \(x\) that satisfies the polynomial equation. Calculators handle this operation efficiently, providing a numerical approximation. In practice, computing \(\sqrt[5]{121.5}\) gives \(x \approx 2.9\). This value completes our solution to the polynomial equation. Underpinning the importance of the fifth root is understanding that it allows us to inverse the fifth power operation applied to \(x\), isolating the variable effectively.